Microrheology of complex ﬂuids - QLIlight.ece.illinois.edu/ECE564/microrheology_review.pdf · of the microrheology of complex ﬂuids are reviewed. ... (MacKintosh and Schmidt 1999).

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 68 (2005) 685–742 doi:10.1088/0034-4885/68/3/R04

Microrheology of complex fluids

T A Waigh

Polymers and Complex Fluids, Department of Physics and Astronomy, University of Leeds,Leeds, LS2 9JT, UK

E-mail: [email protected]

Received 11 October 2004, in final form 16 December 2004Published 9 February 2005Online at stacks.iop.org/RoPP/68/685

Abstract

The field of microrheology is concerned with how materials store and dissipate mechanicalenergy as a function of length scale. Recent developments in the theory and instrumentationof the microrheology of complex fluids are reviewed. Equal emphasis is given to the physicalphenomena probed, advances in instrumentation, and specific experimental systems in whichthis field has already had an impact. The inversion of the compliance data, measurement ofsample heterogeneity, high frequency viscoelasticity, effects of shear flow, single moleculeexperiments, surface viscoelasticity and time evolution studies are considered. The techniqueshighlighted include particle tracking microrheology, diffusing wave spectroscopy, laser track-ing, magnetic tweezers and atomic force microscopy. Specific examples of complex fluidsystems are chosen from the fields of polymers, colloids and biological assemblies.

(Some figures in this article are in colour only in the electronic version)

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686 T A Waigh

Contents

Page1. Introduction 687

1.1. Description of the field 6871.2. Scope of the techniques 688

2. Physical phenomena 6912.1. Introduction 6912.2. Generalized Stokes–Einstein equation 6942.3. Heterogeneity 6962.4. High frequency viscoelasticity 7002.5. Geometry 7012.6. Shear flow 7022.7. Single molecule experiments 7042.8. Surface viscoelasticity 7052.9. Time evolution 707

3. Instrumentation 7083.1. Types of measurement 7083.2. Techniques 712

4. Specific experimental systems 7284.1. Polymers 7284.2. Colloids 7344.3. Biological assemblies 734

5. Future avenues of research 7356. Conclusions 736

Acknowledgments 736References 737

Microrheology of complex fluids 687

1. Introduction

1.1. Description of the field

Rheology is the study of the viscoelasticity of materials. Microrheology extends this definitionto consider how the dynamic behaviour of these materials changes with length scale. Theseparation of a material into ‘component’ or ‘structure’ is often artificial and morphologiescommonly exist at a whole range of length scales in complex fluids. Thus a rich rangeof theoretical and experimental questions are presented to the prospective microrheologist(MacKintosh and Schmidt 1999).

Microrheology relates to the phenomena involved in the storage and dissipation ofmechanical energy in soft materials at the micrometre or sub-micrometre level (MacKintoshand Schmidt 1999). A technical hurdle for optical methods of mechanical spectroscopy istypically set at the micrometre level, which dictates the prefix in the title. There has alsohistorically been a distinct biological bias in the field, since cells operate at the micrometrelevel, providing a strong impetus to drive the research (Berg 1993).

Practically microrheology has long been identified with the resolution limit of an opticalmicroscope operating at its largest degree of magnification, ∼0.5 µm, and optical techniquescontinue to dominate the field. However, the area is rapidly enlarging and advancing with aseries of technical and theoretical barriers being overcome. Within this enlargement there hasbeen the creation of new sub-fields which are the subject of more specialist reviews; magnetictweezers (Strick et al 2003), force microscopy (Mukhopadhyay and Granick 2001), particletracking (Tseng et al 2002a, 2002b), surface viscoelasticity (Meyer et al 1998), microfluidics(Hansen and Quake 2003) and diffusing wave spectroscopy (DWS) (Harden and Viasnoff2001). In this paper the basic universal principles of the techniques are discussed and moremodern developments are reviewed.

The field of complex fluids is an emerging area of physics. It coexists with suchconventional areas as condensed matter and fluid mechanics. There is an extremely wide rangeof sample chemistries, topologies and geometries that are investigated in complex fluids, inmany ways richer and more demanding than those included under more traditional physicalclassifications. In particular, complex fluids (polymers, colloids and biological materials)are expected to demonstrate behaviour intermediate between solids (completely elastic) andfluids (completely viscous), and accurate methods are thus required to quantify the phenomenaassociated with their viscoelasticity. The industrial applications of complex fluids are vast.How do you choose the viscosity of a new shampoo in a new formulation, a drug deliverysystem as it is transported through the body, a fruit cake mixture as it experiences multiplephase transitions during cooking or a grade of cement in the construction of a new building(Larson 1999)? There is thus a wide range of fields that will benefit from new rheologicalprobes driving the current research.

Microrheology is closely connected to the field of microfluidics, which considers suchphenomena as those involved in ink jet printing, microelectrophoresis on a chip, microvalves,and the kinetics of protein crystallization (Hansen and Quake 2003). The change of emphasis,which separates the two fields, is the extension to the consideration of viscoelasticity withmicrorheology. The overlap is thus quite strong and the fluid mechanics of materials in confinedgeometries is a common area to the two research fields.

The subject of biological motors has recently been revolutionized by new microrheologytechniques (Howard 2001). The understanding of the viscoelasticity of the motors at themolecular level has been improved, which directly relates to questions of muscular dysfunctionvitally important to the medical industry. Intensive research funded from studies into

688 T A Waigh

cardiac disease with optical/magnetic tweezers and atomic force microscopy (AFM) has beentransferred to more general areas of complex fluids research.

There is a long history of colloidal probe particles (Mhetar and Archer 1996) being usedas tracers to define flow kinetics in engineering studies of bulk viscoelastic materials. Herea number of assumptions are made; the probe particles are isokinetic with the motion of thematerial investigated, there are no interparticle interactions involved between the probes, andthere is no restructuring of the viscoelastic material by the probes (Solomon and Lu 2001). Theexperimental situation with these tracer studies is thus much more complicated than typicallyfound with particle tracking microrheology (PTM) and many of these assumptions need to berevisited if engineers wish to quantify their measurements in more detail.

1.2. Scope of the techniques

Historically the roots of microrheology can be found in the observation of the Scottish biologist,Robert Brown (1827) that pollen grains (his probe particles) moved incessantly on the surfaceof water. This phenomenon at first sight appears abnormal; what is the origin of the forcedriving the motion? Such behaviour was later theoretically analysed by Albert Einstein(1905), who established the molecular nature of matter by explaining Brown’s results in termsof a statistical analysis of the collisions of pollen with the surrounding solvent molecules.The calculations of Einstein were subsequently shown to be quantitatively correct by thepainstaking experiments of Jean Baptiste Perrin (1948). Perrin demonstrated that the mean-square-displacement (MSD) of 0.37 µm gutta-percha particles in water is directly proportionalto time with a constant of proportionality that describes the frictional dissipation of the particles.Another historical land mark is seen in early attempts to create active magnetic microrheometersto manipulate micrometre-sized ferromagnetic particles (Crick and Hughes 1950), althoughthe first suggestion of such a device is much earlier (Seifriz 1924). After a period of relativeinactivity, there followed modern developments such as particle tracking video-microrheology(Mason et al 1997), AFM (Mahaffy et al 2000), DWS (Pine et al 1988) and optical tweezers(Starrs and Bartlett 2003). These methods have principally been driven by the availability ofcheap computing power and intense coherent monochromatic light sources (lasers).

In the present day, the case for microrheology as a new analytical technique for complexfluids is very strong. For the simplest available technique, PTM, the apparatus is relativelycheap (<£10 k), and sufficiently simple that internationally competitive apparatus can beadapted and constructed without advanced technical ability. Some of the specific advantages ofmicrorheological methods will be enumerated to make a clear case for the range of techniquesthat are reviewed and to emphasise some unifying features:

(1) Combinatorial chemistry. Microrheology allows the rapid characterization of microlitrequantities of material, enabling detailed phase diagrams to be quickly established (Amisand Schubert 2004, Breedveld and Pine 2003). An order of magnitude increase inthroughput has recently been demonstrated with particle tracking of viscoelastic peptides,when compared with previous bulk rheology measurements. Further increases inthroughput are possible, that will scale with improved computing speed in the case oftracking experiments (Breedveld and Pine 2003). Microrheology presents an ideal methodfor probing permutations of reactants during the synthesis of a new viscoelastic material.

(2) Agreement with bulk values. Measurements obtained on expensive low shear rate solutionrheometers have been shown to agree with those taken on budget microrheology equipmentwith a subset of complex fluids (see section 2.3). A consensus on how the sample chemistryand heterogeneity relate to the question of micro/bulk agreement needs to be constructed


in full, but there are good prospects that the range of overlap can be improved usingtwo-particle cross-correlation techniques (Crocker et al 2000).

(3) High frequency response. The low inertia of colloidal probe particles allows themeasurement of the high frequency response of soft materials with methods such as DWS(Mason and Weitz 1995) and optical tweezers (Schnurr et al 1997). There are no prospectsfor such large frequency windows being developed with conventional bulk rheometers dueto the large forces and torques required.

(4) Minimization of sample volumes. Reducing the amount of viscoelastic material necessaryto perform a measurement of a complete viscoelastic spectrum causes a concurrentreduction in cost and increase in the flexibility with which the sample geometry can bechosen. For example, the characterization of specimens from medical trials necessitatesthe use of minute histology samples and microrheology could provide a diagnostic methodin these cases.

(5) Length scale. The viscoelasticity can be characterized as a function of length scale withinthe sample. Complex fluids typically exhibit hierarchical structures on a series of separatelength scales and measurements at each scale would enable a complete picture of theirviscoelasticity to be developed. Specifically, techniques such as particle tracking video-microscopy, optical/magnetic tweezers and AFM, allow the viscoelasticity to be quantifiedas it varies from point to point (Tseng and Wirtz 2001).

(6) Sizing of particles. This can be performed using microrheological measurements andoffers a method for characterizing the molecular structure of a complex fluid. The intrinsicviscosity of the solution is plotted against the particle concentration and subsequentanalysis can provide a radius of gyration for the particles (Goodman et al 2002). Thisprocess could compete with other standard methods of particle characterization such aslight scattering techniques (Chu 1991) and bulk viscometry (Kulicke and Clasen 2004),since it requires much smaller sample volumes, i.e. microlitres compared with millilitres.

(7) Phase diagrams. Measuring the viscoelasticity rapidly at a wide range of concentrationsallows phase diagrams to be mapped. Modern scaling theories allow these phase diagramsto be motivated for a range of complex fluids and microrheology offers a relatively fastroute for their determination. Investigations are not restricted to the sample chemistry;new insights into the physics are also possible, e.g. the dynamic modes that controlthe viscoelasticity of a polymer solution: Zimm, Rousse, sticky Rousse, or reptation(Rubinstein and Colby 2003).

(8) Cellular function. Microrheological techniques provide new methods for probingintracellular function. Many cellular processes involve changes in the viscoelasticityof biomaterials at the micron level such as phagocytosis and motility. Microprobes can beintroduced into living cells to quantify their motility without completely disrupting theirfunction (Lau et al 2003).

(9) Delicate probes. Thermal energies (∼kT ) on colloidal particles provide a delicate probe ofthe structure and dynamics of fragile soft condensed matter systems. Materials which aredamaged during measurement on bulk rheometers often fare better with microrheologicaltechniques, which typically apply forces on the order of a few piconewtons. It is thenpossible to be certain that measurements are in the linear rheological regime.

(10) Single molecule experiments. The theoretical analysis of the fluctuation spectra ofmicrometre sized probe particles relates directly to a range of single molecule forcespectroscopy techniques. For example, microrheology experiments are required tocalibrate optical/magnetic tweezers and atomic force microscopes in such studies.Furthermore a second generation of single molecule experiments are investigating thedynamics of soft condensed matter systems on an individual molecular basis. Rheological

690 T A Waigh

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Figure 1. Typical ranges of the (a) frequency and (b) shear modulus (G′, G′′) calculated inmicrorheology techniques during measurement of the linear viscoelasticity of complex fluids. Theresponse at low moduli has not been investigated in detail except with video particle tracking.

concepts are required to understand the response of these viscoelastic specimens (Okajimaet al 2004).

Before entering into a discussion of specific experimental techniques, it is useful to tryto get a broader outline of the range of frequencies and moduli that can be measured. Forexample, how is it possible to measure the linear viscoelastic spectrum of a new specimenof a complex fluid (figure 1)? Here there is a sharp dichotomy between passive and activemicrorheological techniques, where the probe particles are subjected to thermal energy or an


external field, respectively. Passive techniques are typically more useful for measuring lowvalues of predominantly viscous moduli, whereas active techniques can extend the measurablerange to samples containing significant amounts of elasticity.

It is instructive to compare microrheology with the field of inelastic scattering, which is astandard method for studying the wide range of dynamic processes that occur in complex fluids.Inelastic and quasi-inelastic scattering of light (Berne and Pecora 2000), x-rays (Grubel andZontone 2004) and neutrons (Higgins and Benoit 1994) provide complementary informationto rheological studies. The common bond between microrheology and scattering is that bothallow non-invasive measurements of the time response of fragile complex fluids. This is not truewith many electron/x-ray/confocal microscopy, and bulk rheological methods. Furthermoreboth microrheology and scattering offer a length scale sensitive measure of the dynamics ofcomplex fluids. Microrheology is starting to compete on this front with inelastic scatteringtechniques, in contrast to previous bulk rheological methods, which typically provide timeaveraged measurements over many millilitres of material. Rheological measurements havethe advantage that they quantify how materials store and dissipate energy, and thus directlyrelate to every day macroscopic observables, whereas scattering methods are connected withless physically tangible intensities in reciprocal space.

2. Physical phenomena

2.1. Introduction

It is useful to consider some of the general principles of the physics which underpin themicrorheological techniques described. A rich variety of phenomena are observed in lowReynolds number dynamics. The phenomena have has been investigated in detail in molecularbiophysics with respect to the motility of cells and microorganisms. The reader is directedto two clear and insightful pedagogic accounts (Berg 1993, Purcell 1977) for an introduction.The low Reynolds number approximation to the dynamics of small colloidal particles can oftenbe invoked in microrheology experiments and can be crucial to provide tractable data analysis.

The Reynolds number for the motion of a small particle in a fluid (�, inertial force/frictionforce) is given by:

� = νRρm

η, (1)

where η is the viscosity of the fluid, ν is the particle velocity, R is the particle radius and ρm isthe relative density. This dimensionless group (�) determines when it is possible to neglect theinertial force terms (m dv/dt ∼= 0) in a particular system. From a basic analysis of Newton’ssecond law for a particle moving in a viscoelastic material (mass m), the simplified equationof motion (no viscoelastic memory) is

mdv(t)

dt= FSto(t) − γ

dx(t)

dt− κx(t), (2)

where Fsto(t) is the stochastic thermal force acting on the system, γ is the drag coefficient, κ isthe elastic constant, and x(t) is the displacement of the particle as a function of time. It can beshown that in most passive microrheological experiments the viscous forces of the surroundingmedium dominate the dynamics of the probe particles and the particles will coast to a halt ina few Angstroms if there are no forces actively driving their motion (Berg 1993). Althoughthe average particle displacement of the probes is zero, they are constantly fluctuating due tothermalized collisions with surrounding solvent molecules, 〈x2(t)〉 �= 0.

The buoyancy of the probe particles is another important practical factor to be consideredin making accurate rheological measurements. The particle dynamics need to be examined

692 T A Waigh

unmodified by the effects of sedimentation, far away from the perturbing effects of boundaries,to perform meaningful measurements. For example, probe particles dropping into and out of theplane of detection in particle tracking experiments can introduce high levels of low frequencynoise drastically reducing the quality of measurements.

Using Archimedes principle, the effective mass (m′) of a particle suspended in a fluid isgiven by its actual mass (m) minus the mass of the fluid it displaces (particle volume (V )×fluid density (ρ)).

m′ = m − Vρ. (3)

The number density of the particles N(z), as a function of their height (z) above thebottom of the container, is given by a Boltzmann distribution when the particles are in thermalequilibrium (Berg 1993).

N(z)

N(0)= e−m′gz/kT , (4)

where z is the height of the particles, m′ is the mass adjusted for the buoyancy, N(0) is theparticle density when z is 0, g is the acceleration due to gravity and kT is the thermal energyscale. The concentration of probe particles in a container will thus change exponentially withheight in thermal equilibrium.

The velocity (vsed) adopted by the particles as they sediment also needs to be considered,i.e. the rate at which they approach equilibrium (Berg 1993). A simple analysis of the equationof motion of a non-interacting single particle in a purely viscous fluid gives an expression forthe sedimentation velocity

vsed = 2a2m′g9V η

, (5)

where a is the particle radius, η is viscosity, g is gravity, and V is the particle volume.For example, a polystyrene sphere of 1 µm radius and specific gravity ρs = 1.05 g cm−3

(iron oxide particles have densities as high as 5.2 g cm−3) will sediment in water with a velocity(vsed) of 1.1 × 10−5cm s−1. It will take about 2 h for such spheres to sediment 1 mm makingtracking experiments in water possible. In contrast iron oxide particles require only 1 minto travel this distance making passive techniques difficult with such probes. It is a tacitassumption that low rates of sedimentation (vsed) do not affect the fluctuation spectra measuredin the perpendicular directions in PTM experiments and thus dense particles can be used formeasurements in highly viscous materials. There continue to be a series of theoretical questionsin the hydrodynamics of sedimentation that are not well understood (Dufresne et al 2000).

The question of buoyancy in PTM sets a number of experimental challenges; how are theprobe particles to be prevented from dropping out of solution or how can the experiment beperformed before this happens? One solution is to increase the viscoelasticity of the materialsexamined, but this can cause problems if the probe fluctuations are reduced below a detectablelevel. An ideal solution is when the sedimentation can be adjusted with an applied potentialin situ with active microrheology techniques, such as optical and magnetic tweezers (Gosseand Croquette 2002, Starrs and Bartlett 2003). An optical or magnetic force can thus be usedto balance the gravitational force.

Diffusion is a process whose conceptual complexity is often underestimated by a superficialexamination of the basic mathematics involved. The diffusion coefficient in two dimensions(calculated, for example, by a time series of digitized camera images) of a particle in a purelyviscous fluid allows the amplitude of particle fluctuations to be quantified.

The mean square fluctuation of a particle in n dimensions (Berg 1993) (〈r2〉) dependslinearly on time (t) with a proportionality constant D defined to be the diffusion coefficient


for translational motion

〈r2〉 = 2nDt (6)

and 〈r2〉 = 〈r2x 〉 + 〈r2

y 〉 in two dimensions (n = 2), the sum of the mean square fluctuations inthe x and y components of the MSD.

The linear time dependence 〈r2〉 ∼ tα , with α equal to 1 in equation (6), corresponds tothe case of diffusion in a purely viscous material, e.g. water or glycerol. Introducing an elasticelement in the complex fluid causes the value of the exponent to reduce at short times andsub-diffusion is observed, α < 1. Quantitative analysis of this subdiffusive motion can allowthe calculation of the rheological properties of the material and this is the key concept behindmany of the microrheological techniques that will be discussed (section 2.2).

The diffusion coefficient (D) of a probe particle (units cm2 s−1) can be calculated from thefluctuation dissipation theorem in its simplest form i.e. it is inversely related to the frictionalcoefficient (f ) of the particle scaled by the thermal energy (kT )

D = kT

f, (7)

where f for a colloidal sphere can be calculated from Navier–Stokes equations and is given by:

f = 6πηrh (8)

and thus the Stokes–Einstein relationship can be formed by combining equations (7) and (8)

D = KT

6πηrh, (9)

where η is the viscosity of water, rh is the hydrodynamic radius and kT is the thermal energy.Rotational diffusion can also be considered in a similar fashion to translational motion to

study the response of a complex fluid. In this case the fluctuations in the rotational displacementof a particle in three dimensions are given by the formula

〈θ2〉 = 6Dθt, (10)

where 〈θ2〉 is the mean square fluctuations in the angle of rotation of the particle, Dθ is therotational diffusion coefficient (units s−1) and t is the time.

For a rigid sphere with stick boundary conditions, the rotational diffusion coefficient (Dθ)

can be calculated by the fluctuation dissipation theorem equation (7), where in this case therotational frictional coefficient (f ) is 8πηr3

h .

Dθ = kT

8πηr3h

, (11)

where η is the viscosity, and kT is the thermal energy. The dependence of the rotationalfluctations on time become sublinear with the introduction of an elastic component in thematerial, in a similar manner to translational fluctuations.

The fluctuation spectra of minute particles embedded in a complex fluid are thus seento be a measure of its viscoelasticity. The next challenge covered in section 2.2 is how tobridge the gap from the microscopic behaviour to macroscopic measures of viscoelasticity(G′, G′′, J (t), etc). The relevant measures of bulk viscoelasticity are described in detail insection 3.1. Although the emphasis is often made in microrheology on the measurementof shear moduli (G′, G′′), it is also possible to examine longitudinal moduli (E′, E′′) withmicrodynamic mechanical testing apparatus (µDMTA), i.e. AFM.

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2.2. Generalized Stokes–Einstein equation

A recurrent question in the analysis of the fluctuation spectrum of particle displacement inpassive microrheology experiments is how to transfer between the compliance (proportionalto the mean square amplitude of the particle fluctuations, equation (34)) as a function of timeand the linear viscoelastic spectra (shear modulus) as a function of frequency. Measurement ofa wide time range is required to provide the corresponding width in frequency space. A numberof different methods are indicated in the literature for calculating this transformation and it isa well-defined class of mathematical ‘inverse problem’, having general features in commonwith such classic techniques as the Fourier transform (Bracewell 1986).

Historically the generalized Stokes–Einstein (GSE) equation was initially put forward onan ad hoc basis to analyse the thermal fluctuation spectrum of probe spheres (Mason and Weitz1995). It was later placed on firmer theoretical foundations (Levine and Lubensky 2000). Thealgebraic form resembles that of the Stokes–Einstein equation (9), although it now involvesLaplace transformed quantities

D̃(s) = kT

6πasη̃s

, (12)

where a is the radius of the probe sphere, η̃s is the Laplace transformed frequency dependentviscosity, D̃(s) is the Laplace transformed frequency dependent diffusion coefficient and s isthe Laplace frequency.

The GSE equation applied to colloidal hydrodynamics is thought to be valid if the inertialeffects of the probe particles, the inertial effects of the fluid and the longitudinal compressionmode of the fluid can all be neglected over the frequency range of the measurements. Theeffect of the particle inertia is shown to be negligible if the frequency obeys the inequality(Levine and Lubensky 2000)

ω �(

9G(ω)

2a2ρf

)1/2

, (13)

where G(ω) is the frequency dependent shear modulus, a is the particle radius, and ρf is thedensity of the particle. This introduces an upper bound on the frequency of a measurement,on the order of 10 MHz in a typical system (polystyrene spheres of 0.2 µm diameter inpolyethylene oxide (PEO) solutions).

Similarly, the fluid inertia can be neglected if the measurement frequency obeys theinequality

ω �(

π2G(ω)

4a2ρf

)1/2

. (14)

This frequency is again on the order of 10 MHz.The longitudinal compressive modes are typically on the order of 10 Hz for a polymeric

material providing the lower frequency limit of the measurements. Thus there is a widefrequency range over which the GSE can be used with accuracy for colloidal hydrodynamicsin practical situations (10–107 Hz).

Although they are in principle equivalent for well-defined MSD data sets, it is useful tomake a comparison of the different inversion methods, which are required in order to transferbetween the shear moduli and the compliance. Figure 2 shows a graphical representation ofthe possible methods of data inversion. Fourier transform methods (power spectral density〈r2(ω)〉) may be useful for data sets over a wide range of frequencies with a wide range offeatures (Schnurr et al 1997). However for the limited data sets typically encountered inpractice with microrheology experiments the numerical method using analytic continuation is

Gabi

Pencil


<r2(s)>

<r2(t)>Measured

MSD

Mason’snumericalform (1)

UnilateralLaplace

Transform

)(~

sG

G(t)Stress relaxation

modulus

Inverse Laplace transform or relaxation

spectrum (2)

G*(ω)Complex

shear modulus

Unilateral Fourier Transform

<r2(ω)>

Power Spectral Density

(Bracewell1986)

Kronig-Kramersand fluctuation

dissipation theory (Schnurr et al 1997)

Analyticcontinuation

(Masonet al 1996)

GSE

Figure 2. Possible methods of data inversion to provide the complex viscoelastic shear moduli(G′, G′′) from the compliance (or mean-square particle fluctuations 〈r2(t)〉). s is the complexLaplace frequency, and ω is the experimental frequency.

(1) Mason and Weitz’s numerical form for the shear modulus as a function of frequency(Mason 2000)

G(ω) ≈ 2kT

3πa〈r2(τ )〉�(1 + (d ln〈r2(τ )〉/d lnτ ))

∣∣∣∣∣τ=1/ω

,

where kT is the thermal energy, a is the probe radius, r2(τ ) is the MSD calculated at frequencytime τ = 1/ω and � is the gamma function.(2) The shear modulus (G(s)) as a function of the Laplace frequency (s) can be fit to thefunctional form ∑

j

(Gj s

s + 1/τj

),

G(t) is reconstructed as the sum of exponentials amplitude Gj and time constant τj .

becoming a standard tool (Mason et al 1997), since the Fourier transform method can providenoisy data with spurious high frequency fluctuations (Schnurr et al 1997). The Provencheralgorithm is a readily available robust implementation of a numerical inverse Laplace transform,to construct the stress relaxation modulus from G̃(s) (equation (12)), but analytic continuationis a faster tool for smooth data sets with few distinct features (Mason 2000).

The GSE equation in terms of the Laplace-transformed modulus G̃(s) takes the form

G̃(s) = sη̃(s) = s

6πa

[6kT

s2〈r2(s)〉 − ms

], (15)

where m is the particle mass, and r2(s) is the Laplace-transformed MSD. The additionalsubtracted term ms is included for completeness as an inertial correction. In the limit ofa freely diffusing particle at low frequencies in a purely viscous fluid there is a simplifiedexpression for the Laplace-transformed MSD (Mason 2000)

〈r2(s)〉 = 6D

s2(16)

and the frequency independent viscosity is recovered η = kT /6πaD equivalent to the Stokes–Einstein equation (9).

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Table 1. Classifications of homogeneous and heterogeneous microrheology systems typicallyencountered. Materials can transform from homogeneous to heterogeneous as a function ofconcentration, e.g. DNA in dilute solutions is effectively homogeneous (Goodman et al 2002)and heterogeneous in semi-dilute solutions (Chen et al 2003).

Homogeneous (1 point microrheology measurementstypically give agreement with bulk rheology)

Heterogeneous (viscoelasticity of the material variesdramatically from point to point)

Neutral polymers in good/theta solvents, e.g. polystyrenein toluene/decalin (Starrs and Bartlett 2003), PEO inwater (Mason et al 1997, Schnurr et al 1997)

Chemically cross-linked polymeric gels (charged,uncharged), e.g. actin with cross-linker proteins,poly(vinyl alcohol) (Narita et al 2001)

Charged flexible polymers in good/theta/bad solvents, e.g.polyacrylic acid (good), polystyrene sulphonate (bad),titin (theta) (Di Cola et al 2004), hyaluronic acid (good)

Physically cross-linked biopolymer gels, e.g.carragenans, pectins, collagens (Velegol and Lanni2001), starches (Heinemann et al 2004)

Charged semi-flexible polymers, e.g. DNA (Goodman et al2002), actin (Xu et al 1998), myosin, de novo peptide(Aggeli et al 2001)

Associating polymers, e.g. ionomers,hydrophobically modified polyelectrolytes(Di Cola et al 2004), hydrophobic/hydrophilicblock copolymers (Lu and Solomon 2002)

Molecular liquids, e.g. water (Crocker and Grier 1996),glycerol, ethylene glycol

Gelled and jammed colloids, e.g. cheese, cement,jammed micelles

Colloidal fluids, e.g. PMMA spheres, polystyrene spheres(Sohn et al 2004), tomato bushy stunt virus, silica particles

Biological assemblies, e.g. cells (Fabry et al 2001,Lau et al 2003)

Lyotropic liquid crystals, e.g. surfactants (Cardinaux et al2002), tobacco mosaic virus, actin

Theoretical questions have been raised concerning the applicability of the generalizedStokes–Einstein equation (12) to charged sphere suspensions. Due to the long-range natureof electrostatic interactions, a charged tracer sphere experiences particularly strongly thediscontinuous nature of its environment which could cause a break down of the GSE equation(Nagele 2003). However experimentally good agreement between micro and macrorheologyhas been found for highly charged linear polyelectrolytes with single PTM (section 4.1.2), sofurther experimental and theoretical work is required for a range of different charged complexfluids to properly understand this behaviour.

The analysis of the unconstrained motions of fluctuating particles has been considered inthis section, i.e. passive microrheology techniques. The fluctuations in the motion of a trappedcolloid (magnetic/optical tweezers) require consideration using the Langevin equation (36)and are discussed in section 3.2.3.

2.3. Heterogeneity

Many materials contain heterogeneous/inhomogeneous structures in nature, i.e. they arestructured on a range of length scales greater than that of their molecular arrangement atthe Angstrom scale. This phenomenon provides a challenge for the microrheologist, sincethe heterogeneities are typically on the length scale of the size of the probe particles and theycan be the dominant factor determining the result of a viscoelastic measurement (Levine andLubensky 2000).

Practically, the question of heterogeneity relates to an important problem; is the responseof one probe particle a true measure of the bulk rheology? Forces between probe particlescan sensitively affect their dynamics and a wide range of mesoscopic forces are possible withcomplex fluids (Evans and Wennerstron 1994).

Table 1 includes a list of materials, which are typically homogeneous or heterogeneousat the micron level with microrheology experiments. A familiar example of a heterogeneous


ξ

ξI

Polystyrenechains

Denserregions

Figure 3. Schematic arrangement of polystyrene chains in a good solvent deduced from smallangle neutron scattering measurements. Two length scales, ξ the mesh size and ξI the size of theheterogeneities are required to fully describe the system (Bastide and Candau 1996, de Luca et al2004).

(a) (b)

Figure 4. (a) Mean-square fluctuations of the particle displacement of colloidal probes embeddedin partially cross linked actin, (b) histogram of the incidence (counts) of particle fluctuations of aparticular amplitude for a time lag of 0.1 s (inset shows time lag 1 s) (Tseng and Wirtz 2001).

material is table jelly (jelo), in which relatively fluid sections (low number of collagen crosslinks) are interspersed with dense elastic regions (high number of crosslinks) at the microscopiclevel (Velegol and Lanni 2001). However, heterogeneous viscoelasticity is not specific to gelsystems, any complex fluid in which the size of the probe particle is of the length scale ofinhomogeneities in its structure, could cause the material to dissipate energy differently frompoint to point.

Even with some ‘standard homogeneous fluids’, as defined in table 1, such as polystyrenein a good solvent, two correlation lengths (ξ and ξI) are sometimes required to describe staticneutron scattering experiments. There are heterogeneities on two simultaneous scales, the10 nm length scale and the nanometre scale, even with this well-characterized, well-behavedcomplex fluid system, see figure 3 (Medjahdi et al 1991).

A challenge with PTM is encountered in the study of associating polymers which ispractically closely connected with the examination of heterogeneity. This relates to thestabilization of the associating system against flocculation when mixed with the probespheres (Lu and Solomon 2002, Valentine et al 2004). Flocculation would halt any seriousmicrorheological investigation and is the extreme limit of strong probe/fluid attraction withthe sphere perturbing the mesh.

Histogram methods are a useful tool for quantifying the heterogeneity in the compliancefrom multiple one-particle tracking experiments, in which the magnitude of particle fluctuationsfor a particular time step is plotted against the number of particles in an ensemble experiencingthe fluctuations (figure 4). Large fluctuations correspond to less dense regions of the polymeric

698 T A Waigh

Figure 5. (a) Trajectory of a trapped fluctuating particle, (b) y coordinate of (a) as a functionof time, (c) jumps between trapped trajectories over long times, and (d) y coordinate of (c) as afunction of time (Wong et al 2004).

network and vice versa. There is an open theoretical question on how to relate the statisticaldistribution of the particle fluctuations to both the mesostructure and bulk rheology of a sample(Tseng and Wirtz 2001).

Further questions related to multiple particle tracking experiments in polymeric mesheshave been examined (Wong et al 2004). The importance of the ratio of the probe particleradius (a) to the mesh size (ξ) in determining the results of one-particle microrheologyexperiments was demonstrated. When the size of the particle is on the order of the mesh size(a ∼ ξ) anomalous subdiffusive dynamics were observed in the tracked particle trajectoriesdue to probe spheres jumping between cages created by the actin mesh (figure 5). Thus forboth one and two-particle experiments using materials containing large mesh sizes, care mustbe taken to account for such cage hopping artefacts. The statistics of the histograms of suchparticle motions (figure 4) have not yet been quantitatively related to the statistical mechanicsof the activated diffusion in a particular system, although the framework of the mathematicsrequired in terms of the probability of a particle jumping between cages has been sketched out(Wong et al 2004).

A new technique that overcomes some of the problems caused by heterogeneity hasrecently been introduced using the cross correlation of the motion of two thermally fluctuatingprobe particles (Crocker et al 2000, Levine and Lubensky 2000). Through examinination ofthe viscoelasticity of guar gels, it was established that the one-particle fluctuations were largerthan the fluctuations measured from cross correlation (figure 6(a)). The linear viscoelasticitycalculated from the one-particle technique largely underestimates the bulk value by a factorof 5. More importantly the cross-correlation method showed good agreement with standardbulk rheology techniques (figure 6(b)) and the method has now been established with a seriesof other complex fluid systems (DNA (Chen et al 2003), living cells (Lau et al 2003), andactin (Crocker et al 2000) etc).

The diffusion coefficient (Drr) for correlated fluctuations of two-particle motions alongthe line connecting them takes the form of a GSE equation (12) rescaled by a factor of 3

Drr(r, s) = kT

2πrsG(s), (17)

Dθθ = Dφφ = 12Drr, (18)

where kT is the thermal energy, r is the particle separation, Dθθ and Dφφ are components of thediffusion tensor corresponding to the transverse components of the two-particle fluctuationsin spherical coordinates.

A demanding test of the analytic machinery to quantify heterogeneity using two-particlecross correlation has been observed in the application of PTM techniques to living cells


Figure 6. Comparison of the data from single-particle and two-particle cross-correlation trackingexperiments with guar; (a) mean square particle fluctuations, triangles correspond to single particleand circles the cross-correlated particles (inset shows the behaviour of two-point correlationfunction versus particle separation where the 1/r behaviour is emphasized, or comparison withequation (18)). (b) The linear viscoelastic shear moduli calculated from the MSDs, circlescorrespond to the two-particle technique and triangles one-particle technique. Continuous linesrefer to bulk measurement of the shear moduli in fair agreement with data from cross-correlatedparticle motions (Crocker et al 2000).

(Lau et al 2003). The internal cellular stress fluctuations were found to have a ω−2 powerspectrum, as expected for a material with a slowly evolving internal stress.

Optical tweezer techniques (Starrs and Bartlett 2003) to measure the rheology ofheterogeneous matter have subsequently been demonstrated using the same idea ofcross correlation between the motion of the particles, but require a slightly differentmathematical formalism. Here the fluctuations of two colloidal particles held in two separateoptical traps are cross correlated and the analysis invokes a Langevin equation (36) with anexplicit trapping potential. The microrheology of the correlated motion of the two trappedspheres is considered in terms of their individual friction coefficients. Newton’s second law isapplied to each particle relating the acceleration to the applied force (Starrs and Bartlett 2003).For example, for the displacement (x1) of particle 1 it is found that

mdu1(t)

dt=

∫ t

−∞ξ11(t − t ′)u1(t

′) dt ′ −∫ t

−∞ξ12(t − t ′)u2(t

′) dt ′ − kx1(t) + f R1 (t), (19)

where f R1 (t) is the stochastic thermal force, m the colloidal mass, u1 and u2 are the velocity

of particles 1 and 2, respectively, k is the spring constant of the optical trap, t is the time,ξ11 and ξ12 are, respectively, the force acting on one moving sphere when the second sphereis stationary and the force generated on one sphere by the motion alone of the secondarysphere. Such optical tweezer techniques can provide the high frequency rheology (∼10 kHz)of heterogeneous materials (Starrs and Bartlett 2003).

Stress relaxation has been used to measure the heterogeneity in living cells by means ofmagnetic probes and active pulsed magnetic fields (Bausch et al 1999). The stress fields can bemapped across the cellular microstructure and then related to its biological function. Similarlythe viscoelasticity across a cell can be mapped using magnetic cytometry by exerting a torqueon ferromagnetic particles (Fabry et al 2001).

The disagreement in one-particle oscillatory magnetic microrheology with bulk rheologyexperiments has been examined (Schmidt et al 2000). It has been demonstrated thatmicrorheology experiments with sinusoidally oscillated magnetic beads underestimate theviscoelasticity of actin solutions.

There is the intriguing possibility of using cross correlation of driven magnetic particlesto study the local rheology of highly viscous samples in a manner similar to two-particle

700 T A Waigh

optical tweezer and particle tracking experiments to provide better agreement with bulkrheology results. Here the cross-correlated motion would need to be calculated betweenactive and passive particle motions, i.e. magnetic beads and surrounding inert probe colloids(Evans 2004). This cross-correlation method holds the prospect of a high modulus rheometerfor heterogeneous complex fluids. A further idea is to examine the cross correlation ofthe rotational motion of rod-shaped particles (Reichert and Stark 2004), e.g. anisotropicferromagnetic colloids experiencing a torque from a magnetic field and passive anisotropicparticles.

Dynamic heterogeneous rheological phenomena such as shear banding (Olmsted 1999)and phase separation (Sohn et al 2004, Tanaka 2000) could be areas for future research oncethe behaviour of static heterogeneous systems have been thoroughly investigated.

2.4. High frequency viscoelasticity

The high frequency mechanical spectroscopy of complex fluids has only recently been subjectto serious examination. The sole bulk technique available is the use of torsional oscillation,which provides measurements up to 10 kHz but only at a series of fixed frequencies (Fritz et al2003), and is consequently not widely used. In comparison high frequency microrheologytechniques, primarily DWS (Mason and Weitz 1995) and optical tweezers (figure 1), offermeasurement of the linear rheology over a continuous range up to megahertz frequencies.There are a number of new novel dynamic processes which have been demonstrated andthere are good prospects for relating the observed high frequency phenomena to the moleculardynamics of the components of the material. The theory to explain the new results is beingdeveloped hand in hand with the experiments and important advances have been made (Morse1998). High frequency microrheology can provide relaxation times for the internal modes(∼10 ns) of individual molecules in solution, which are typically unavailable from conventionalsolution-state bulk rheology (Xu et al 1998).

The crucial facet of the new microrheological measurements is that the small inertia ofthe probe particles facilitates the measurements, i.e. the probe particle motion can be rapidlyreversed as required with high frequency oscillatory readings. Rapid reversal of the appliedtorque is not possible within conventional bulk rheometers, which typically have an upperfrequency limit of 100 Hz for measurements of continuous spectra.

The inertia of the solvent at high frequencies with solution state complex fluids is observedas a correction to the dissipative shear modulus G′′ − ωη (Ferry 1980, Larson 1999, Massaet al 1971) (ω is the frequency, and η is the viscosity of the solvent). A breakdown of thestandard assumptions of the coarse-grained nature of the frictional coefficient of the solventis thought to occur at still higher frequencies (above megahertz frequencies) (Morris et al1988). This breakdown has, as yet, not been observed in microrheology techniques up to10 MHz frequencies. The experimental evidence is from oscillatory electrical birefringenceexperiments that indicate that the rotational motion of solvents associated with polymersdepends on their concentration (Lodge 1993).

Measurements of non-linear rheology have been achieved using DWS on colloids undershear (Uhomoibhi and Earnshaw 2000). Interpretation of the correlation functions from theexperiments is still at an early stage of development.

Predictions exist for the high frequency viscoelasticity of polymers for free-drainingand non-free-draining flexible chains in solutions, but as yet have only been well tested forintermediate frequencies (Ferry 1980). Claims have been made for the novel behaviour ofthe high frequency viscoelasticity of PEO in water, but this needs to be verified by furtherexperiment (van Zanten et al 2004).


� (rad/s)

G�,

G�

(dyn

es/c

m2 )

Figure 7. The linear viscoelasticity of semi-dilute semi-flexible polymers (actin). The highfrequency ω3/4 modes for the shear moduli are shown for both the elastic (G′) and dissipative(G′′) components (Xu et al 2002). Dots indicate DWS data and filled diamonds are from bulkrheology.

Semi-flexible polymers are a testing ground for microrheological methods (figure 7).Further, as yet unseen, ultra-fast dynamic modes (less than nanoseconds) are now predictedfor the high frequency fluctuations of semi-flexible polymers (Liverpool and Maggs 2001).These timescales may well be overlapping with those at which non-coarse-grained specificsolvent/polymer interactions occur (Lodge 1993).

The GSE relation (equation (12)) has been tested with hard sphere colloids at highfrequencies and the relevant dimensionless groups have been examined (Sohn et al 2004).The GSE equation works very well in this case.

The development of the double-trap optical-tweezer cross-correlation technique allowsthe measurement of the high frequency viscoelasticity of heterogeneous systems as describedin section 2.2. DWS in contrast is only able to measure the ensemble averaged one-particleresponse (figure 7).

2.5. Geometry

The field of microfluidics has demonstrated many phenomena in which the geometry of afluid system relates directly to the application, e.g. electrophoresis on templated structures(Hansen and Quake 2003), dying textile fibres (Quere 1999) and ink jet printing (Hansen andQuake 2003, Probstein 1994). To date there have only been a small number of microrheologyexperiments in confined geometries, but there is a large scope for the extension of the methods,driven by the possible applications.

The hydrodynamic interaction between two isolated colloids has been measured usingin-line optical tweezers (Bartlett et al 2001) (figure 8). The hydrodynamic interaction witha surface has been studied (Dufresne et al 2000). These optical tweezer methods also allowaccurate measurements of interparticle potentials in confined geometries, although experimentsare limited to optically transparent materials.

Depletion (Verma et al 2000), bridging (Kampf et al 2004) and steric forces (Meyer et al1998) are intimately related to the geometry of the confinement of a complex fluid and will havea direct impact on microrheology measurements. Silica spheres were found to have a depletionattraction in optical tweezers experiments when placed in DNA solutions in agreement withthe Asakura/Oosawa model (Verma et al 2000).

There is a wide range of dynamic physiochemical effects, which are related to surfacetension, such as coating flows and the Rayleigh instability (Probstein 1994). Such behavioursare only just starting to attract attention from the microrheological community. The changein viscoelasticity across a liquid–liquid interface has been considered (Sohn et al 2004). This

702 T A Waigh

Distance to

the surface

Optical traps

Dielectric

spheres

Figure 8. The hydrodynamic interaction between two colloidal spheres near to a surface can bemeasured using in line optical tweezers (Dufresne et al 2000).

Figure 9. The degree of extension of single DNA molecules both above and below the overlapconcentration (c∗) from fluorescence microscopy studies. The Weissenberg number (Wi) isincluded for each concentration (Hur et al 2001).

elegant study uses single optical fibre DWS to measure the rheology in a phase-separatedpolymer mixture.

2.6. Shear flow

Only a few studies on microrheology in shear flow have thus far been reported. They are directanalogues of bulk measurements, i.e. the complex viscosity (η∗) is calculated as a function ofthe shear rate (γ̇ ).

Single DNA molecules have been examined in shear flow (Hur et al 2001, Larson et al1997, LeDuc et al 1999, Perkins et al 1997, Smith et al 1996). DNA dynamics were probedas a function of the Weissenberg number (Hur et al 2001); the ratio of the diffusive timescale (tD) of the translational motion of the chain to the characteristic flow time scale (1/γ̇ ).Furthermore, the degree of extension of the DNA molecules could be measured as a functionof strain (figure 9) (Hur et al 2001). An additional study by this group demonstrated thebehaviour of chains in elongational flows using a crossed-slot shear-flow geometry (Perkinset al 1997). Another fluorescent DNA experiment combined with an optical trap was used totest a non-linear elastic dumbbell model for the chain hydrodynamics (Larson et al 1997).


Light source DNA chain Coverslip

Piezoelectric

motor control

SlideInverted

fluorescence

microscope

Buffer

solution

Figure 10. Dynamics of single biological macromolecules under shear were investigated by movinga coverslip over a microscopy slide controlled with a piezoelectric motor. The fluorescing tagsattached to the macromolecule are observed with an inverted microscope containing a suitablelight source (LeDuc et al 1999).

Magnetic

beadSemi-dilute

actin mesh

Direction of

directed

reptation

Tethered

actin filament

Figure 11. Actin filaments can be forced through a semi-dilute mesh using magnetic microrheology(Dichtl and Sackmann 2002).

Another simple, experimentally elegant method was designed to measure the response ofa single DNA chain (LeDuc et al 1999). Here a DNA chain was attached to a microscope slideand the corresponding coverslip was glued to a piezoelectric motor stage as shown in figure 10.The dynamics of the chains under shear could be followed by imaging a fluorescent tag onthe molecules using an inverted microscope with a suitable light source, such as a correctlyfiltered mercury lamp or a laser.

A sophisticated method to measure motion in shear flow uses magnetic tweezers (Dichtland Sackmann 2002). Forced reptation of semi-flexible actin molecules in semi-dilutesolutions was examined using this method (Dichtl and Sackmann 2002) (figure 11). Theforce on a single actin fibre can be studied and the viscoelastic relaxation of the motion ofthe fibre is represented using an equivalent circuit (figure 12). Anomalously high valuesof the frictional coefficient of the longitudinal motion of the fibres were found. They werenot compatible with predictions from simple reptation theory and more work is needed tounderstand the measurements. Such experiments are subject to many of the same questionsconcerning the effects of microheterogeneity (section 2.3) as standard thermally activated one-particle tracking measurements. For example, how are the probe particle fluctuations activatedby their hopping motions through the matrix of a heterogeneous complex fluid?

Optical tweezers are more limited for shear flow experiments, since the optical traptypically cannot be moved over such a wide length range limiting the applied stress.Furthermore the small trapping forces available with optical tweezers provide low measurableshear rates (see section 3.2.3) even in low viscosity materials such as water. DWS experimentshave been applied to sheared colloidal motion, but the data analysis still offers a number ofintriguing questions (Hebraud et al 1997, Uhomoibhi and Earnshaw 2000).

704 T A Waigh

Figure 12. The stress response as a function of time of a single actin fibre experiencing forcedreptation followed by relaxation. The equivalent circuit used to model the data is shown as an insetwith two viscous dashpots and an elastic element (Dichtl and Sackmann 2002).

Perpendicular

fluctuations

Laser

Surface

Oligomer

Physical/chemical

tether

Chemical

tether

Cantilever

Figure 13. Schematic arrangement of an AFM single molecule microrheology experiment(MacKintosh and Schmidt 1999). The viscoelasticity of a single oligomer can be examined tetheredbetween a surface and the cantilever.

2.7. Single molecule experiments

Following the success of single molecule force studies to examine tensile properties with AFM,and optical/magnetic tweezers, an obvious extension is to measure the dynamic modes of thespecimens. With AFM this research is at a fairly early stage of development with questionsexisting on the analysis of data as a function of the cantilever geometry and the analysis of thestatistics of the motion of a material on a molecule by molecule basis.

As described in the previous section, fluorescent labels attached to single molecules aretypically used to image their dynamics. These fluorescent methods tend to be restricted to thestudy of giant biopolymers such as DNA (LeDuc et al 1999), titin (Tskhovrebova and Trinick2002) and actin (Dichtl and Sackmann 2002). Similarly, optical and magnetic tweezers aretypically restricted to the motions of large molecules (DNA, actin, titin, etc) and in this respectAFM has an advantage; it can be used to study oligomers (figure 13).

With AFM, optical- and magnetic-tweezer single-molecule experiments, an importantquestion is whether the chemistry used to tether the molecules to the surface/probe affects theparticle dynamics. A large amount of effort is thus expended in preparing protocols for correctsample adhesion.

In AFM, single molecule studies are often statistical in nature. An ensemble of moleculesare stretched one after another using an automatic motorized routine. The experimentalistneeds to decide which scans are characteristic of a molecule and which contain artefacts due to


the attachment of more than one molecule, the failure of cohesion or impurities. Attempts atdata analysis with dynamic AFM on oligomers tend to be model dependent, i.e. the fluctuationspectrum of a cantilever is calculated from a molecular model.

The dynamics of partially stretched protein molecules were examined with a speciallyadapted AFM in which both the molecule and the surface could be sinusoidally oscillated(Okajima et al 2004). This allowed the viscoelasticity of the proteins to be examined duringstretching.

Optical tweezer experiments (Svoboda and Block 1994) can be used in either probe/surfaceor probe/probe geometry (in-line tweezers) enabling complicated surface interactions to beavoided.

Cheap magnetic tweezers have enabled parallelized measurements on the single molecularelasticity of DNA chains (Assi et al 2002). This could allow combinatorial chemistry onthe single molecule level (perhaps with biotechnological sequencing applications) or theexamination of a large number of high modulus specimens (Amis and Schubert 2004).

Theory with regard to the structure and dynamic response of single polymeric chains hastaken some recent advances (Dobrynin et al 1995, Farge and Maggs 1993). The behaviour ofpolymeric chains has been modelled in detail including the effects of both chirality (Morozand Nelson 1998) and semi-flexibility (Liverpool and Maggs 2001).

2.8. Surface viscoelasticity

The dynamics of complex fluids often change dramatically when they are confined near asurface (Meyer et al 1998). The interfacial permutations of gas, liquid and solid interfaces (i.e.gas/liquid etc) all require individually optimized methods for the measurement of the surfaceviscoelasticity.

Both active and passive microrheological techniques (MacKintosh and Schmidt 1999) arepossible to probe the viscoelasticity of a surface. Particles can be embedded in a surface andtheir thermally generated motility quantified using particle tracking microscopy techniques(Saxton and Jacobson 1997). Alternatively magnetic or optical particles can be attached to theinterface and their dynamics probed using magnetic or optical tweezers, respectively.

For liquid/liquid interfaces, the viscoelasticity of membranes has been probed using single-trap optical tweezers (Helfer et al 2001). A large range of interactions are observed withcytoskeletal components (figure 14) and the wide range of possible viscoelastic responses areof direct relevance to the biological function of the membrane. Predictions for the in-plane andout-of-plane fluctuations of the membrane motions were made, related to the complex shearmodulus of the material and compared with experiment.

Particle tracking applications in membrane dynamics (liquid/liquid interfaces) have beenreviewed (Saxton and Jacobson 1997). The diverse range of interactions of bilayers withcytoskeletal proteins on the subdiffusive dynamics of tracked particles have been highlighted.

Liquid/solid interfaces were studied using the interaction of optically trapped spheres ina liquid as they approach a solid surface (Dufresne et al 2000). Corrections were found to thehydrodynamics of the trapped sphere depending on the distance from the surface.

Quartz crystal microbalances are used to probe the viscoelasticity of thin films adsorbed onquartz (liquid/solid interfaces). This is a resonance technique and as such is confined to a seriesof fixed frequencies for rheological measurements, although high frequency measurements canbe performed (Buckin and Kudryashov 2001).

With AFM, the attachment of a colloidal sphere (figure 15) to the cantilever of themicroscope provides a mechanism to simplify the probe geometry, better define its chemistryand thus provide more accurate measurement of the hydrodynamic interaction between the

706 T A Waigh

Membrane

Laser beam

Trapped

colloidal

particle

Out of plane

fluctuations

Figure 14. High frequency fluctuations of membranes can be measured using optical tweezerswith a single trapped colloidal sphere (Helfer et al 2001).

Spherical

Probe

Laser

Cantilever

Viscoelastic

surface

Perpendicular

fluctuations

Figure 15. Out of plane fluctuations of the surface of a complex fluid can be measured with anatomic microscope using a colloidal probe attached to the cantilever (Mahaffy et al 2000).

surface and the probe. In this case the Hertzian approximation can then be invoked to calculatethe viscoelasticity (Mahaffy et al 2000). This would seem to be the most flexible approachfor the analysis of surface viscoelasticity, since it provides direct measurement of the complexlongitudinal modulus (E′, E′′). These rheological functions can be subsequently modelledwith standard rheological theory and compared with bulk measurements. However, the case ofAFM microrheology is still far from being well developed. Problems exist with the tractabilityof calculations with regard to the surface forces and, in much the same way that particletracking experiments are affected by bead chemistry, so too will cantilever, bead and surfacechemistry affect dynamic AFM. The possibility of using unmodified tips as indenters is coveredin section 3.2.5 (Alcaraz et al 2003, Benmouna and Johannsmann 2004).

Tribology, the study of the frictional properties of surfaces (typically solid/solid orsolid/liquid/solid interfaces), is a field that could profit greatly from new microrheologicalmethods (Meyer et al 1998, Scherge and Gorb 2001). Classic problems such as the frictionalproperties of cartilage (driven by medical questions concerning osteoarthritis) are hamperedby the non-planarity of natural samples on the millimetre length scale. Nanotribologymeasurements using flexible probe geometries could revolutionize this area of researchallowing frictional properties to be correlated with diseased states of the material. However,current microtribological methods are often not well-defined physically. The normal force isoften not accurately measured, which is important to calculate friction coefficients (Amontonslaw defines the frictional coefficient (µ) to be the ratio of the frictional force to the normalforce F/N = µ) and care must be taken during analysis with this additional parameter. Surfaceforce apparatus (SFA) can provide accurate values of the friction coefficient in constant(Kampf et al 2004) or oscillatory shear mode (Mukhopadhyay and Granick 2001), but these


Mica cylinders

Reflected light

Complex

fluid

SpectrometerStrain

gauge

Piezoelectric

slider

Figure 16. SFA can be used to measure surface microrheology by oscillating crossed mica cylinders(Boschkova et al 2001). The relative displacement of the surface is measured using ellipsometry.

measurements are for large areas of complex fluids adsorbed on to plane mica surfaces, limitingthe range of problems that can be studied.

SFA can measure the frictional properties of fluids confined between two solid surfaces(solid/liquid/solid interfaces (Meyer et al 1998)), figure 16. Interesting surface effects havebeen described for the rheology of polymer melts. The polymer melts were found to becomeglassy below a critical film thickness due to the pinning of the chains on the mica surfaces(Luengo et al 1997). Another SFA study examined the shear moduli of confined soap films(G′, G′′) as a function of cylinder separation in the range 1–100 Hz. The films were found tohave a dominant elastic component in contrast to their behaviour in the bulk (Boschkova et al2001). With SFA the high frequency limit is again set by the large inertia of the apparatus,limiting the range of measurements.

The rheology of gas/liquid interfaces has been studied using a rotating magnetic rodrheometer (Bantchev and Schwartz 2003, Brooks et al 1999). The diameter of the rods istypically ∼100 µm, on the border of the microrheological regime. There is nothing limitingthe size of the magnetic rod other than the available torque and, if scaled down further witha compensating increase in the sensitivity of the measurement of the particle deflection, thiscould provide a useful method for studying interfacial microrheology.

The nanorheology of a single perfluoropolyether meniscus bridge from a polymericmaterial was examined (Choi and Kato 2003) using two 20 µm glass spheres. The shear moduliwere measured for bridges of length in the range 0–100 nm, at frequencies from 1 to 1000 Hz.

Practically, in most cases experimental studies of surface microviscoelasticity should beconsidered an order of magnitude more difficult than bulk microrheological studies. Howeversuch measurements are of fundamental importance to a series of fields, including membranedynamics, phase separation, adhesion and tribology.

2.9. Time evolution

Complex fluids have a memory, but this memory can change over time i.e. the viscoelasticitycan evolve with time (Ferry 1980, Larson 1999). For example, this could be in a gelationprocess (cooling a gelatine/water mixture), a nematic–isotropic phase transition (such as in theliquid crystalline display of a computer) or a biochemical process in a cell (e.g. contractionof muscle cells in the arm). To measure these processes places a further restriction on amicrorheological measurement; a full spectrum must be acquired in a short time period, toprovide detailed information on the evolution of the viscoelasticity of the material. This feathas been achieved in a multiple particle tracking study (Tseng et al 2002a, 2002b). The timeevolution of the viscoelasticity of F-actin combined with cross-linking proteins was measured

708 T A Waigh

(a) (b)

0.011

10

100

1000 70

60

3 µM

24 µM

10 µM

50

40

30

20

10

00 50 100 150

Gelation time (min)

τ = 0.1 s

200 2500.1

Time lag τ (s)

24 µm

3 µm

1

Ens

embl

e av

erag

e co

mpl

ianc

e (m

2 /N)

Ens

embl

e av

erag

ed Γ

(m

2 /N)

Figure 17. The ensemble averaged network compliance from PTM of actin gels. (a) Time lagdependence of the mean compliance (�), after the initiation of actin polymerization. (b) Evolutionof the ensemble averaged compliance evaluated at a time lag of 0.1 s, during gelation of the solution(Tseng et al 2002a, 2002b).

with an 8 min time step (figure 17). This novel study indicates that it is the degree of networkhomogenization which controls the rate of gelation of F-actin networks (Tseng et al 2002a,2002b). Thus the heterogeneity of the network reduces over time as the system approachesthermal equilibrium.

Phagocytosis (Feneberg et al 2001), the process by which a cell envelopes and transportsfood particles, has been studied dynamically with PTM. A cellular slime mould was examinedas it interacted with folate covered polystyrene beads. Three phases of phagocytosis werefound each of duration ∼10 s. The time resolution for the capture of an individual spectrumwas of the order of 2 s.

Experiments with molecular motors also examine time evolution processes. Myosins (inchworm molecular motors) interact for well controlled time steps as they power themselves alongactin filaments. Both the time steps and lengths of the working strokes have been measuredfor the interaction of single actin fibres with myosin using sophisticated dual-beam opticaltweezers (Rief et al 2000).

The gelation of charged polystyrene colloids destabilized by the addition of salt wasexamined using DWS (Romer et al 2001). The averaging of non-ergodic correlation functionsmeasured from the colloids was considered to obtain the high frequency moduli from theDWS experiments (figure 18). Practically, DWS is limited to sampling rates on the order of∼1 min−1 to obtain well-defined correlation functions, which depend on sample absorption,laser power and detector efficiency. In principle, DWS can be used to investigate the evolutionof features in the high frequency viscoelasticity of complex fluids at this rate.

3. Instrumentation

3.1. Types of measurement

Some standard types of bulk rheological measurement have their analogues in microrheolog-ical techniques (Goodwin and Hughes 2000). It is possible to calculate a number of measuresrelating to the linear viscoelasticity of a complex fluid including G′, G′′ (shear moduli), E′, E′′

(longitudinal moduli), J ′, J ′′ (complex compliance) and the Poisson ratio (ν) with suitableapparatus.

3.1.1. Linear viscoelasticity. A standard linear rheology experiment is to apply a sinusoidalstress/strain and measure the corresponding response of the material through its corresponding


Figure 18. Evolution of the elastic and dissipative shear moduli (G′, G′′) from DWS as a functionof time from the point that the charged colloids are destabilized. The inset shows the long timebehaviour (Romer et al 2001).

strain/stress (Goodwin and Hughes 2000). For example, a shear strain (e) can be applied to acomplex fluid as a function of time (t)

e = Re(e0 exp[iωt]), (20)

where e0 is the strain amplitude, and ω is the applied frequency.The corresponding induced stress (σ ) oscillates in time with a frequency ω, but is offset

by a phase lag (δ)

σ = Re(σ0 exp[i(ωt + δ)]), (21)

where σ0 is a constant stress amplitude in Pascals.The complex shear modulus G∗ in Pascals is then defined by

G∗ = σ

e= σ0

e0eiδ = σ0

e0(cos δ + i sin δ). (22)

The real and imaginary parts of the shear modulus can then be considered separately.The storage modulus (real part) is a measure of the elastic energy stored in the system at

a particular frequency and is given by

G′ = σ0

e0cos δ. (23)

Similarly, the loss modulus (imaginary part) is a measure of the energy dissipated as a functionof frequency:

G′′ = σ0

e0sin δ. (24)

Active magnetic microrheological techniques typically follow the method of applicationof a stress followed by consideration of the resultant strain (the analogue of a bulk stresscontrolled rheometer). A strain controlled device has not yet been built, although it could bepossible with particle tracking feedback control with magnetic/optical tweezers.

A method of comparing the storage and loss modulus is made by the calculation of theloss angle (δ), combining equation (23) and (24)

tan δ = G′

G′′ . (25)

710 T A Waigh

The viscosity is equal to the zero frequency limit of

η = G′′

ωω → 0, (26)

which often is equal to the zero shear rate limit of the bulk viscosity measured at a series ofconstant shear rates:

η = η(γ̇ ) γ̇ → 0. (27)

The real creep compliance (J (t)) can be defined in terms of an incremental longitudinalstrain (e(t)) and the resultant stress (σ).

J (t) = e(t)

σ. (28)

In terms of bead deflections (xd(t)) in a microrheology experiment, bead radius (a) andan applied force (f (t)) the compliance can be calculated as

xd(t) = J (t)f (t)

6πa. (29)

The complex creep compliance (J ∗) can be defined with an oscillatory strain and resultantoscillatory shear stress. The complex compliance J ∗ is defined by

J ∗ = J1 − iJ2 = 1

G∗ . (30)

The real Young’s modulus for an elastic material is given by E

E = σ

e, (31)

where σ is the applied longitudinal stress and e is the resultant longitudinal strain. It can beextended to a complex modulus as a measure of the viscoelasticity in much the same way asthe shear modulus (G∗) and is measured in AFM microrheology experiments.

The Poisson’s ratio (ν) provides information on how a material changes shape whenstrained (Lau et al 2003). For an isotropic material the ratio is given by

ν = −eperp

e, (32)

where eperp is the linear strain in a direction perpendicular to the tensile stress producing thetensile strain e. The negative sign insures that the quantity is positive in most experimentalsituations. The Poisson ratio can be measured for viscoelastic specimens using two-particlecross correlation (Lau et al 2003).

The real shear modulus (G) for an isotropic elastic maerial is G = σc/θ when the shearangle (θ) is very small and σc is a constant stress. The different elastic moduli are interrelatedfor a simple elastic material (Bower 2002).

G = E

2(1 + ν). (33)

It is possible to mathematically transform between different complex measures of thelinear viscoelasticity such as J ∗, G∗, E∗ and the reader is referred to standard texts for theirinterrelationship (Ferry 1980, Goodwin and Hughes 2000).


Cover glass

Magnetic particle

Electromagnet

Pole piece

Viscoelasticspecimen

(b)

Bead

Viscous components (dashpot)

Elasticcomponents (spring)

(c)

(a)

Figure 19. Stress relaxation experiments on the viscoelasticity of a material can be performed by theapplication of a stress field followed by measurement of the resultant strain response. (a) Schematicdiagram of a single pole piece and magnetic probe (b) the applied magnetic field and the particledisplacement in response and (c) the equivalent viscoelastic model (Bausch et al 1999).

3.1.2. Stress relaxation. Active microrheology techniques allow the option of measuringstress relaxation of a complex fluid under the influence of an external force. An example isthe use of magnetic particles tethered to the surface of cells or embedded within them. Theparticles are subjected to a step-like magnetic field and their position with time is measured afterthe application of the force (figure 19). This is a less technically challenging microrheologyexperiment than that for probing the complex shear modulus, because the measurement of thephase information (δ) is not required. Only the measurement of the displacement of the particlein a correctly calibrated magnetic field is required. With a series of magnetic particles it isthen possible to map out the stress relaxation across micrometre-sized samples using an opticalmicroscope (section 2.2). Subsequently the data can be analysed by constructing an equivalentviscoelastic model (Maxwell, Kelvin, standard linear solid, etc) (Bower 2002, Goodwin andHughes 2000) to fit the displacement relaxation observed (figure 19(b)). Such experiments arenot constrained to step pulses, more complex stress functions could be envisaged whose formcould be chosen to fit the application. Correct application of the stress fields requires linearamplification of computer generated signals and negligible inductance of the electromagnetcoils (see section 3.2.4).

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The stress relaxation of single molecules is considered in section 2.6 when actin fibresare subjected to forced reptation. Stress relaxation experiments are also possible using opticaltweezers (section 3.2.3).

3.2. Techniques

A graphical comparison of the range of available apparatus to measure the linear rheology ofcomplex fluids was made in figure 1. In the following section the practical requirements andlimitations of these studies are considered, highlighting the major microrheological methodscurrently available. The techniques described require a degree of sample transparency to makemicrorheological measurements inside the materials.

3.2.1. Particle tracking microrheology. Particle tracking microrheology in its most basicform is a direct continuation from the oldest experimental microrheological study (e.g. Perrin).Probe particles are embedded in a viscoelastic specimen and the rheological characteristicsare extracted from the thermal-fluctuation spectra measured using digital video-microscopy.Present-day studies tend to consider averaging over a large ensemble of simultaneously fluctu-ating particles embedded in a material, since this allows for better statistics on the subsequentrheological functions and facilitates analysis of the sample heterogeneity (Tseng and Wirtz2001). The size of the probe particles is limited in standard video microscopy to opticalwavelengths (∼0.5 µm), as this is the diffraction limited resolution of an optical microscope.

Fluorescent particles can be used to decrease the probe size (∼0.1 µm). This increases thespatial resolution in heterogeneity studies and increases the size of particle fluctuations (∼1/r).Fluorescent probe particles allow viscoelasticity to be probed in complex optical objects, e.g.allowing them to be easily located within living cells. However fluorescence studies canencounter challenges with respect to the chemistry of the fluorescent tags. Photobleachinggives a finite lifetime for the experiments and care is needed to avoid irrevocably changing theparticle dynamics through the attachment of the probes. The measurable dynamic range in allthese studies is defined by both the rate of data acquisition of the digital camera and the speedof data storage using either a CD writer or magnetic memory. It is typically of the order of50 Hz (Chen et al 2003).

Anisotropic probes have not as yet experienced extensive consideration, one exceptionbeing the chains formed from magnetic nanocolloids (∼10 nm radius) (Wilhelm et al 2003).

It is important to quantify the degree of noise inherent in particle tracking experiments,since this defines the largest moduli (smallest amplitude of particle fluctuations) that can bemeasured. Empirically it has been found that the tracked displacement of an immobilizedcolloidal sphere (e.g. glued to a coverslip) underestimates the errors that are included intracked motions (Papagiannopoulos et al 2005). Superior calibration of the particle trackingexperiments can be achieved with respect to the viscosity of a range of standard Newtonianfluids, e.g. glycerol/water mixtures at a range of concentrations. Figure 20 shows a typical rangeof fluctuation spectra measured as a function of time. Measurements with this experimentalset-up (100× oil immersion lens, DVD recorder, Olympus BH2 microscope and Haitsu camera)are limited to glycerol concentrations below 80% glycerol or erroneous subdiffusive artefactsare introduced. Absolute calibration of the displacements in fluctuation spectra are typicallyperformed by imaging a micrometer gauge onto the video microscope. An alternative methoduses the displacement of a piezoelectric crystal and can improve the resolution to ±1 nm inspecific applications.

A related question is the accuracy with which the centre of the particles can be determined.Large particles subtend more pixels on the video camera and the centre of optical mass can


0.1 1

1E-3

0.01

0.1

1

0.5 µm PS-beads

slope = 0.57

slope = 0.94

slope = 1

water/glycerol (w/w)

90/10 80/20 70/30 60/40 40/60 20/80 10/90 0/100

time (s)

MS

D (

µm2)

Figure 20. Mean square fluctuations (MSD) of 0.5 µm diameter polystyrene beads in glycerolmixtures of various compositions allow video PTM experiments to be calibrated (Papagiannopouloset al 2005).

be obtained more accurately. Optical images of small probes are diffraction limited and stepsneed to be taken to deconvolve the diffraction effects. The accuracy with which the opticalcentre of a probe particle can be measured has been studied (Crocker and Grier 1996). It wascalculated that image analysis techniques provide a resolution of ±10 nm in the focal planeand ±150 nm in depth with a 100× oil immersion lens, S-VHS recorder, NEC TI-324A CCDcamera and Olympus IMT2 inverted microscope.

The use of oil immersion optics is invaluable in providing high quality particle trackingmovies free from the effects of particle/surface interaction, due to the increased workingdistance of the lens. Cavity slides (20 µl) can be purchased cheaply and the main limitationfor minimization of the sample volume is that the fluctuations are analysed at a distance fromthe coverslip larger than that provided by Flaxen’s law (Svoboda and Block 1994).

Software for PTM has been written on a number of platforms; IDL (http://glinda.lrsm.upenn.edu/∼weeks/idl/tracking.html), Labview and NIH have been the most popular. Multipleparticle tracking and cross-correlation techniques are computationally intensive and have as yetonly been implemented in IDL. The generic algorithms require the manipulation of large arraysand linked lists for the tracked particle motions, which require more advanced programmingskills than that for single bead motion.

The dependence of the microrheological measurements on the colloidal probe size andon its chemistry need to be carefully considered; phenomena such as mesh hopping (Wonget al 2004) and depletion flocculation are not uncommon (Verma et al 2000). Thus in muchthe same way as a bulk rheologist considers their experimental geometry with relation to theirmeasurements (Couette, cone and plate, etc (Goodwin and Hughes 2000)), a microrheologistmust study the interactions of their probe bead with the complex fluids under examination.

Ideally, the bead chemistry should have a negligible influence on the result of a rheologicalmeasurement. A microrheologist needs to check that the probe particles are stable at the pH at

714 T A Waigh

Particle

trajectory

x

y

(a) (b) (c)

Find

fluctuation

spectrum

<r2> as a

function of

time

Track particles

using image

analysis

software

t

<r2>

ω

G�,G�

Calculate linear

viscoelasticity

G�,G� as a

function of

frequency (ω)

Probe

particle

Figure 21. Particle tracking microrheology enables the linear viscoelasticity of low modulusmaterials to be extracted from the fluctuation spectrum. (a) Trajectory of the probe particle ismeasured, (b) the average fluctuation spectrum as a function of time t is calculated, and (c) thelinear viscoelasticity as a function of frequency ω can then be found.

which the experiments are conducted and that there is negligible adsorption of the complex fluidon the beads surface, e.g. negatively charged fluids require negatively charged probes. Somework has already been published on sterically stabilized probes in organic solvents, but thisrequires careful choice of the complex fluid system limiting the range of applicability (Starrsand Bartlett 2003). Biological samples are restricted to aqueous chemistries, which leads tothe standard use of pH sensitive charged colloids (polyacids, polybases). Experimentalists alsorequire that there are no depletion effects destabilizing the probe particle/specimen mixtures.

A recent study has highlighted the effects of bead chemistry on measurements withbiopolymers (Valentine et al 2004). Probe colloids were incubated with solutions of globularproteins (bovine serum albumin) to reduce adsorption of actin and fibrin. Interestingly theuse of cross correlation between two particles reduces the effects of the bead chemistry on thelinear rheology that is calculated.

This emphasis on bead chemistry can be considered an asset. Too many materials areplaced into bulk rheometers with little thought for the molecular interaction between thesample and the cell. For example, depletion of the material at the walls of a rheological celland dissolution of ionic impurities from the metals in the cell wall are both common artefactswith bulk rheology experiments on aqueous solutions. Particle tracking microrheology allowssuch interactions to be accurately measured.

The experimental steps for calculating the linear viscoelasticity using particle tracking arespelled out in figure 21. Particle trajectories are tracked, the mean-square amplitude of thesefluctuations are calculated and the data is then transformed between rheological functions(typically the compliance is transformed into the complex shear modulus, section 2.2). Ithas been theoretically shown that the MSD of a probe particle (radius a, thermal energy kT ) isdirectly proportional to the creep compliance (J (t)) (Xu et al 1998). The MSD data can thusbe directly converted by rescaling into a standard rheological measure

〈r2(t)〉 = kT

πaJ (t). (34)

It is a challenge to find neutrally buoyant probe particles for experiments in non-aqueous solvents. Polystyrene/toluene solutions were studied using PMMA tracers, and


Probe colloid

mg

Vρgz

Lateral 2-D

fluctuations

Figure 22. The sedimentation of a probe colloid in a viscoelastic solution (Berg 1993). At lowReynolds number it is assumed that the lateral fluctuations in particle displacement are decoupledfrom the sedimentation velocity.

good agreement was found with accepted rheological parameters at intermediate semi-diluteconcentrations. However, sedimentation was a problem limiting the concentration range thatcould be measured and buoyant probes were not readily available. The terminal sedimentationvelocity is a concern for PTM (figure 22). Problems are accentuated with complex fluidssuspended in organic solvents whose density is often less than water (m′ is large in equation (5)).Therefore the probes rapidly sediment, and make measurements of low modulus materials verydifficult.

A major advantage of magnetic microrheology is that a vertical component of the magneticforce can offset the weight of the particle providing neutral buoyancy (section 3.2.4). Thequestion of buoyancy is an important concern for measurement of low viscosity materialswith dense metallic probes. Similarly optical tweezers can trap sedimenting probe particles,facilitating measurements with sedimenting probes (Starrs and Bartlett 2003).

Cross correlation of particle motions offers a new method for measuring sampleheterogeneity. An experimental drawback with two-particle tracking experiments is that theyrequire careful handling of the image analysis of long movies (∼30 min) and extremely well-defined sample environments to reduce high noise levels when compared with one-particleexperiments (Crocker et al 2000).

The technique of particle tracking has been used to probe the dynamics of membranesystems. These allow the kinetics of reactions among membrane species to be measured(Saxton and Jacobson 1997).

3.2.2. Diffusing wave spectroscopy. The origins of DWS can be traced back to the seminalwork of Weitz and Pine (Pine et al 1988). It was realized that the technique of photon correlationspectroscopy could be extended into the multiple scattering limit and there were a number oftractable methods of data analysis allowing the structure and dynamics of novel materials tobe probed, e.g. foam coarsening (Hebraud et al 1997), and the aggregation process in cheesemaking (Horne 1989). Many of these subjects could not be analysed using standard lightscattering methods, since the single scattering approximation is invalidated with these opaquematerials (Stepanek 1993).

Subsequently, the DWS method was extended to become a new microrheology techniqueusing multiple scattering from monodisperse colloidal probe particles embedded in atransparent viscoelastic material (Mason and Weitz 1995) (figure 23). This provides both theautomatic ensemble averaging of a scattering technique and access to the ultra fast dynamicsresolvable with commercially available photon correlator electronics (10 ns). Fast dynamics ina correlation function from dynamic light scattering (DLS) correspond to high frequenciesin the complex shear moduli (∼106 Hz) when the data are mathematically transformed.

716 T A Waigh

Laser

Photodiodecorrelator(backscattering)

Photodiodecorrelator(transmission)

Viscoelasticmaterial

Colloidalprobe

Rectangularcuvette

Polarizers

Figure 23. DWS can be used to measure the high frequency linear viscoelasticity of complex fluidsusing a laser with high frequency stability, probe particles in a transparent viscoelastic matrix anda fast correlator in transmission or back scattering geometries.

Non–ergodicsample

Referencecell

Input laser

Photodiode andcorrelator

L1 L2

Figure 24. A reference cell placed in front of the non-ergodic sample of interest allows thecorrelation function to be physically averaged facilitating data analysis (Scheffold et al 2001).

Physically, the colloidal probes provide a strong contrast mechanism for light scattering.The large refractive index of the spheres masks the effects of the weakly scattering (non-contrast-matched) viscoelastic background, and establishes the accuracy of the mathematicalapproximations required for analysis of the correlation functions.

Back scattering and transmission geometry microrheology experiments are now fairlyroutine (Mason and Weitz 1995, Xu et al 1998). Practically it is deduced that an experimentis in the multiple scattering limit when the depolarized scattering is of the same magnitude asthe polarized scattering (Ivv = Ivh). This is achieved by rotating a polarizer in front of thephotodiode correlator and increasing the width of the sample cuvette or concentration of theprobe particles until the depolarization condition is fulfilled. The DWS method is available toanyone with DLS equipment, square cuvettes and a flexible goniometer which can be placed ateither 0˚ or ∼170˚. Care is required to attenuate the signal in transmission geometry to avoiddestroying the detector with the direct beam.

Experimentally a challenge to DWS is how to achieve rapid averaging of correlationfunctions over a series of sample volumes, with gelled or glassy non-ergodic specimens.This hurdle has now been overcome using the reference cell method (Scheffold et al 2001)(figure 24). A mathematical theorem (equation (35)) was established, the multiplicationrule, which states that the ensemble averaged field autocorrelation function g

(2)1 (L1, L2, τ )

of the double cell combination of the sample and opaque reference equals a product of theautocorrelation functions g

(1)1 (L1, τ ) and g

(1)1 (L2, τ ) corresponding to the individual cells.

Thus, individual measurement of g(2)1 (L1, L2, τ ) and g

(1)1 (L1, τ ) followed by division provides

the ensemble averaged autocorrelation function of interest g(1)1 (L2, τ ).

g(2)1 (L1, L2, τ ) ≈ g

(1)1 (L1, τ )g

(1)1 (L2, τ ). (35)


(a) (b)

Figure 25. DWS experiments with polystyrene spheres embedded in worm like micellar solutions.(a) Correlation functions from the transmission and back scattering (inset) geometry, and (b) thecomposite compliance calculated for polystyrene spheres with both geometries (Cardinaux et al2002).

The equation holds under certain well-defined circumstances; in particular scattering inthe first reference layer should be much greater than the second sample one and there shouldbe negligible losses of light at the interface. An alternative mechanical solution to obtainensemble averaged correlation functions with non-ergodic samples is to oscillate the locationof the incident laser beam (Nisato et al 2000) achieving averaging over a series of colloidalconfigurations.

CCD detectors with DWS (Cardinaux et al 2002, Furst and Gast 1998) are well matchedto the study of the slow dynamics of glassy complex fluids, since they typically have veryslow read out times and averaging over many pixels provides high resolution data. The backscattering geometry can be used to access slow relaxation processes, since the distributionof multiple scattering paths (length, s) is broad (probability distribution, P(s) ∼ s−3/2). Acombination of CCD detectors and backscattering geometry extends the dynamic range thatcan be measured, figure 25.

DWS techniques have been used with single optical fibres to probe picolitre rheologywith the detection of back scattered light (Popescu et al 2002) (figure 26). Many applicationsare foreseen (microphase separated materials, surface viscoelasticity and viscoelasticity inconfined geometries) if the dependability of the technique could be established and protocolscreated for introducing the probe particles and fibres in a non-invasive manner. An excitingapplication of this picolitre DWS technique has been to measure the viscoelasticity across theboundary of a phase-separated polymer mixture (Sohn et al 2004) .

DWS has been used to examine the dynamics of sheared colloidal suspensions with lightscattering through dense suspensions in a flow cell (Uhomoibhi and Earnshaw 2000). Anadditional study examined the in situ behaviour of the coarsening of Gillette shaving cream(Cohen Addad et al 1998). Experiments on emulsions subject to oscillatory motions have beenstudied using echo DWS (Hebraud et al 1997).

3.2.3. Laser particle tracking (optical tweezers). The fluctuation spectrum of a probe particleembedded in a complex fluid can be measured using the laser light forward-scattered from asingle particle with a quadrant diode. The incident laser can lead to a tweezing action in thecase of high fluxes of light, although this is not necessary for microrheological measurements(Svoboda and Block 1994). Thus laser tracking can lead to both ‘passive’ and ‘active’ typesof microrheology.

718 T A Waigh

40 µm

1×2 Fibre

Optic Coupler

Laser

diode

Spectrum

analyser

Incident light

Reflected

light

Optical fibre

Colloid

Coherence

volume

Viscoelastic

specimen

Computer

analysis

Figure 26. Optical fibre DWS can measure picolitre rheology using back scattered light fromcolloidal probes embedded in a viscoelastic material (Popescu et al 2002).

(a) (b)

Figure 27. The power spectrum of optical tweezers in the frequency domain has a particular formderived from the Langevin equation (36) due to the interplay of the trapping potential and randomthermal forces. (a) The force as a function of particle displacement in the optical tweezers, and(b) the fluctuation spectrum of the trapped sphere (Svoboda and Block 1994).

Quadrant diodes are a critical technology involved in optical tweezers and AFM.Differential photon-induced voltages are measured between pairs of the four diodes allowingaccurate x–y positioning of the particle at high frequencies (∼10 kHz). However, the operationof the noise function in quadrant diodes has only just begun to attract serious attention. It isthought to be a delicate interplay between electronic processing, semiconductor noise, andthe quantum interaction of photons with semiconductors (Berg-Sorensen and Flyvbjerg 2004,Howard 2001).

The use of laser trapping for microrheology experiments complicates the analysis ofmicrorheology data, since both the trapping force and the stochastic collisions with theviscoelastic sample effect the dynamics of the probe particles. A Langevin equation is requiredto analyse the fluctuations of the trapped beads. For low Reynolds number with a Newtonianfluid it is given by

−ηdx

dt+ κx = F(t), (36)

where F(t) is the stochastic thermal force, κ is the elastic constant, and η is the drag coefficient.The particle fluctuations 〈r2〉 now have an upper bound at large times due to the trapping force(or low frequencies in figure 27(b)).


Figure 28. Schematic arrangement of the apparatus required for dual trap optical tweezers forviscoelastic studies. L1 and L2 are two orthogonally polarized laser beams, QD1 and QD2 arequadrant diodes, and PBS is a polarized beam splitter (Starrs and Bartlett 2003).

Optical tweezer apparatus used in force measurement (figure 27(a)), often employmicrorheological techniques to calibrate the strength of the laser traps. Analysis of theBrownian motion of a trapped colloidal sphere can provide the strength of a trap with sub 1%error using statistical analysis of the Langevin equation and well characterized photomultipliertubes (Berg-Sorensen and Flyvbjerg 2004). An alternative calibration method is to push fluidthrough the sample cell containing the optical trap and probe sphere. The trapped particlebecomes dislodged from its optical potential well when the Stokes frictional force, equation (8),becomes equal to the trapping force. Such shear flow experiments (section 2.6) are thuscommonplace with optical tweezers.

Recently in-line dual trap optical tweezers have been used to study the viscoelasticity ofcomplex fluids (Bartlett et al 2001). The traps can be constructed from a single laser dividedusing a beam splitter and their position adjusted in the focus of an optical microscope using amirror. A typical arrangement for a dual beam trap is shown in figure 28 (Starrs and Bartlett2003).

The cornering frequency (f0 on figure 27(b)) measured from the fluctuation spectrum ofa trapped bead provides an accurate method to calibrate optical traps. Theoretically it canbe calculated from a Fourier transform of the Langevin equation, equation (36). The powerspectrum (

∑x) as a function of the frequency (f ) is found to be given by∑

x

(f ) = kT

π2γ (f 20 − f 2)

, (37)

f0 = κx

2πγ, (38)

where kT is the thermal energy scale, γ is the drag coefficient, and κx is the effective lateralspring constant.

Viscous corrections to the coupled motion of two particles near to a flat surface (Dufresneet al 2000) have been measured with dual optical traps. Corrections to the fluctuationspectra of the trapped particles persist to surface/particle separations of tens of particleradii, emphasizing the need for the isolation of particle dynamics away from surfaces toprovide correct microrheological analysis. Flaxen’s law for simple liquid hydrodynamicscan be invoked for one-particle interaction with a surface and this consideration is useful in arange of microrheology techniques (Svoboda and Block 1994). The law requires that the ratio

720 T A Waigh

(A–B)/(A+B)

X

Photodiodes

AnalogProcessing

Lens

Dichroic Mirror

WollastonPrism

Condenser

Specimen

Objective

WollastonPrism

LASER

Bead trapped inoverlapping laserfoci

X

Polarizer/Beamsplitter

λ/4Plate

B

A

Figure 29. Setup of a back focal plane interferometer to measure the linear viscoelasticity ofcomplex fluids (Schnurr et al 1997).

of the distance to the surface (h) to the colloidal diameter (a) is greater than fifty (h/a > 50),for the hydrodynamic corrections in water of the particle motion to be less than 1%. It providesa useful practical limit for probing bulk viscosities in confined complex fluids, e.g. samplescontained in microscope slide cavities.

The fluctuations of membranes associated with cytoskeletal proteins have been studiedusing optical tweezers (Helfer et al 2001). The measured shear moduli for the compositemembranes were consistent with measurements from actin solutions.

DLS experiments have been performed with optically trapped micrometre spheres usinga second laser orthogonally incident to the first (Viana et al 2002). It is not evident whetherthe increased complexity in setting up the coincident lasers is offset by the ease of calibrationof the trap using DLS and the increased dynamic response.

The possibility for tailored optical potentials (saw-tooth functions, etc) through modulationof the trapping laser beam implies that more sophisticated rheological experiments are possible,such as stress relaxation. Oscillating tweezers have been used to extract the storage anddissipative shear moduli of complex fluids (Hough and Ou-Yang 1999). Furthermore, polarizedoptical traps can induce rotation in dielectric spheres. This effect could also be used to exploreviscoelasticity, but only a small torque is provided by the absorption of the angular momentumfrom photons, limiting measurements.

Back focal plane interferometry is an accurate method for measuring particle fluctuationsin laser particle tracking, figure 29 (Allersma et al 1998). The laser beam is split into twooppositely polarized components, and focused on the probe particle. The relative changes inpolarization upon scattering provide information on the particle displacements. The technique


Magnetic pole

pieceGradient field

Constant field

Saturated

paramagnetic

particle DNA chain

Sample cell

Figure 30. One-dimensional magnetic tweezers for force–extension/stress relaxation experiments(Haber and Wirtz 2000). Two sets of electromagnetic coils provide a gradient field over a constantbackground magnetic field.

has the advantage that the displacement detection is biaxial, has higher resolution than thestandard arrangement, and is independent of the position of the laser focus in the specimenplane.

Thermal noise imaging has been achieved in three dimensions by tracking an opticallytrapped sphere with a photonic microscope (Rohrbach et al 2004). The three-dimensionalpositioning of the particle has been measured using both scattered laser light (Rohrbach et al2004) and off-focus imaging of the fluorescent particles (Speidel et al 2003).

3.2.4. Magnetic tweezers. Magnetic tweezers (figure 30) and microrheometers fall into the‘active’ category of microrheological techniques. Relatively large forces (2000 pN) can beapplied to complex fluids extending the measurable range of moduli above that available withpassive particle-tracking methods (the thermal forces are typically sub-pN in value). Indeedviscosity measurements on non-Newtonian fluids have been obtained up to 107 Pa using afalling magnetic needle in a strong magnetic field (Chu and Wang 1992). The frequencyresponse could not be measured with the falling needle technique, the data are output as theviscosity plotted as a function of the shear rate over a limited range (0.007–10 Hz).

Oscillatory magnetic microrheology has the advantage that it can use heterodyne detectionwith video recording techniques, since there is a controllable driving force. Thus the maximumresolvable frequency is set by the speed of the camera shutter (∼1 ms) and the induction ofthe magnetic coil (Fabry et al 2001). It is not the time between subsequent frames (∼10 ms)as with passive techniques (figure 1).

Magnetic tweezers also have the advantage over their optical counterparts that theygenerate no heat in the sample examined, can have a uniform force field over the entire fieldof view (µm2 s) and can orient objects regardless of their geometry (Strick et al 2003). Theydo have the disadvantage that it is difficult to make multiple independent traps.

There is an important choice in the strategy for the construction of magnetic tweezers,which is dependent on the nature of the magnetism in the probe particles. The main division isbetween ferromagnetic and super paramagnetic tweezers. Practically the distinction dependson both the size and chemistry of the particles. Particles whose size is reduced below theWeiss domain of the material are superparamagnetic and those above are ferromagnetic (Trepatet al 2003). Usefully for single molecule experiments, super paramagnetic particles do notexperience a large torque, they rapidly adjust their induced dipole moment to avoid rotationalmotion and this facilitates the data analysis of their translational motion.

The calibration of magnetic tweezers holds some interesting questions, which are solvedby a number of sophisticated methods. The fluctuation spectrum of the spheres in the trapping

722 T A Waigh

potential can be considered in much the same fashion as with optical tweezers, but the strengthof the magnetic field can also be studied in situ from point to point using Hall probes (Gosseand Croquette 2002). A complicating factor is that the magnitude of the induced magneticdipole in the probe can depend on the magnitude of the applied field.

The potential energy (U ) of a magnetic dipole (m) placed in a magnetic field (B) is givenby the scalar product

U = −m · B. (39)

Thus a free permanent dipole experiences a torque as it minimizes the energy by aligning thedipole with the magnetic field.

The corresponding magnetic force (F ) is the gradient of the potential (∇U) and forsuperparamagnetic spheres m ∼= Mmax.

F = −∇(m · B) (40)

F ≈ MmaxVdB

dx, (41)

where V is the particle volume, and Mmax is the saturated magnetization. The torque on alarge ferromagnetic particle (4 µm) can be quite considerable (∼1000 pN µm) and magnetictweezer cytometry has found applications in determining the elasticity of cells adhered tomagnetic beads (Laurent et al 2002).

Magnetic forces can be used to construct tweezers in an analogous manner to their opticalcounterparts through creation of magnetic field gradients in three dimensions (equation (39))(Gosse and Croquette 2002). Accurate methods to calculate the position of the particle inthree dimensions are thus required. A current solution has been to use video imaging for thex–y components of the particle displacement and to analyse the diffraction rings around theparticles to obtain the z component corresponding to the height of the particle. The diffractionring method requires careful preparation of well parallelized illumination optics (Gosse andCroquette 2002). Off-focus imaging with fluorescent particles could equally have been usedin this application (Speidel et al 2003).

Feedback control of the applied magnetic forces is also required to create a direct analogueof optical tweezers, i.e. to hold a particle with a constant force as it is moved across the fieldof view. The feedback control has been achieved electronically by on-line monitoring of themagnetic field and fluctuation analysis of the displacement of the probe particles (Gosse andCroquette 2002).

The frictional force on a particle in a Newtonian fluid can be used to calibrate magnetictweezers if the field strength and dipole moment are both unknown. For example, in the drivenmotion of a superparamagnetic sphere, the probe quickly reaches its terminal velocity oncethe magnetic driving force is balanced by the frictional force (equation (8)). Dipole momentson the magnetic spheres can be separately quantified using a SQUID or a vibrating samplemagnetometer.

A series of different arrangements have been designed to manipulate magnetic particleswith a varying number of pole pieces (1, 2, 4, 6 and 8). There is as yet no consensus as to themost effective set up for rheological measurements.

Single pole piece magnetic microrheology was examined in the context of stress relaxation(section 3.1, figure 19 (Bausch et al 1999)). This is the simplest magnetic rheological apparatusto build, since it does not require measurement of the phase angle of the complex moduli,equation (22). Such a device with a single needle-shaped pole piece was used in a biologicalstudy of the formation of intracellular neurites (Fass and Odde 2003).

A different strategy has been used (Haber and Wirtz 2000) for the micromanipulation ofDNA attached to a magnetic particle using two pole pieces (figure 30). For force extension


Amplifier

Digital image

processing

Light

Magnetic

coils

D/A card

A/D card

Digital image

Coil current

Sample

cuvette

CCD camera

Figure 31. Magnetic microrheometer for studying the linear viscoelasticity of complex fluids(Keller et al 2001). The digital video images and the coil current are acquired simultaneously andstored on the hard disk of the personal computer.

Figure 32. Magnetic tweezers designed for use with ferromagnetic particles (Trepot et al 2003).The air flow is used to control the sample and electromagnet temperature.

studies a constant magnetic field was applied to saturate the magnetic dipole moment of theparamagnetic probe particles and the particle was subsequently moved using a second gradientmagnetic field, equation (41). The arrangement of combined constant and varying magneticfield cannot be used for oscillatory experiments, since the constant field would also need tooscillate, defeating its purpose in saturating the magnetization of the paramagnetic probes.

An oscillatory two pole piece magnetic bead microrheometer has been built which has ahigh frequency response (50 Hz, figure 31) (Keller et al 2001). This frequency response ismade possible by using a fast CCD attached to a frame grabber card synchronized with anadditional A–D card connected to the magnetic pole pieces to measure the phase shift. Theapparatus was used to analyse the scaling behaviour of the linear rheology of actin, with resultsin agreement with the theory for semi-flexible polymer dynamics. Heterodyne detection couldincrease the frequency response by an order of magnitude in such studies (Fabry et al 2001).

Ferromagnetic tweezers with two pole pieces have been demonstrated (Trepot et al 2003).They were designed with large coaxial coils (280 mT, gradient 2T/m, figure 32). Thesetweezers allowed for reasonable forces (∼2 pN) and homogeneous fields (this type of designis used in NMR apparatus), but there is the complication of torques rotating the particles andthe remnent hysteretic magnetic moment of the ferromagnetic particles. The arrangement

724 T A Waigh

Figure 33. Four pole piece magnetic tweezers for particle manipulation in two dimensions(Amblard et al 1996). The split pole pieces are arranged above and below the microscope slideused to contain the sample material.

also suffered from the probe particles being too dense and experiencing problems withsedimentation. The sedimentation phenomena would preclude them being used for anythingbut the most viscoelastic specimens (to decrease vsed in equation (5)). A possible adaptationdeveloped in our laboratory is to place an additional single electromagnetic coil around themicroscope objective to counteract the buoyancy force.

Four pole piece tweezers have been constructed (Amblard et al 1996), in which the fields(and hence forces) were controlled using Hall probes glued to the bottom of the pole pieces(figure 33). Electronic feedback control of the current through the pole pieces was used tomaintain a constant magnetic force on the probe. Both translational (∼1 pN) and rotational(∼10−14 N m torques) manipulation of the motion of the probe was possible with the equipmentand the time resolution was 1

30 s.A six pole piece magnetic tweezer set-up has been reported (Gosse and Croquette 2002).

The arrangement of the apparatus is shown in figure 34. The pole pieces are placed abovethe sample cell and the vertical component of the magnetic field gradient (equation (41))allows positioning of the particles in the z direction (figure 34(a)). A sophisticated feedbackmechanism is used to move the particles both horizontally and vertically. A constant challengewith magnetic probe manipulation is to quantitatively understand the response of the particle,which depends on the magnetic field two fold both through the induced magnetization M , anddirectly on the magnetic field strength (B). Calibration was achieved with the six pole piecetweezers by image analysis of the fluctuation spectrum of the probe particles. This analysis ismathematically similar to the method used with optical tweezers, i.e. the Langevin equation(equation (36)) is analysed (Strick et al 2003).

Eight pole piece tweezers have been built (Huang et al 2002). The arrangement of theeight pole pieces appeared to provide superior magnetic force control than those of Gosseet al although quantitative analysis of the fluctuation spectrum of the tracked particles wasnot made.

A unique study of the viscoelasticity of polymeric materials was made using the dynamicsof a falling magnetic needle (Chu and Wang 1992). The position of the needle (∼millimetresin length) was measured to great accuracy using light scattered onto a position detector. Theneedles were repeatedly magnetically levitated, allowed to sink for a small distance (∼100 µm),repositioned at the initial starting point and the positional data averaged. The falling needleprovided measurement of the viscosity as a function of shear rate for polymer solutions over awide range of concentrations. Perhaps the biggest advantage of this technique is that it providesan extremely large range of measurable viscosities (0.001 Pa < η < 107 Pa), although it doessacrifice the ability to probe heterogeneity by using such large probe particles.


Pole pieces

Lens

SN

z

y

Sample

F

µ

Amplifiers

CCD camera

PC

Video aquisition

LensFocus Control

Oil

Current Control

Electromagnets

(a)

(b)

Figure 34. Three-dimensional magnetic tweezers for particle manipulation (Gosse and Croquette2002). (a) A close up of the arrangement of the pole pieces around the sample, and (b) theglobal arrangement of the apparatus, with feedback control from the computer used to position theparticles.

The study of the rotational motion of ferromagnetic probes has experienced a numberof experimental studies (Laurent et al 2002, Wilhelm et al 2003). An impressive application ofrotational magnetic microrheology (Fabry et al 2001) was made in which the shear modulus ofcells was extracted from the complex torque and angular displacement, equation (47) at highfrequencies.

Another method to study micro viscoelasticity using the rotational motion of probeparticles has been made by means of the response of chains of magnetic colloids (Wilhelmet al 2003). It would require much further development for such a technique to become astandard characterization tool. The polydispersity of the chain lengths provides a complicatedbarrier to the analysis, although the 10 nm particles did have the strong advantage of avoidingthe phagocytosis mechanism in living cells. This facilitates the microrheological analysis ofintact biological organisms.

An interesting idea is to extend cross-correlation techniques (discussed in both the sectionson particle tracking and optical tweezers) to magnetic microrheology. One possible methodis to cross correlate the motion of ‘driver’ magnetic beads with that of passive markerparticles, potentially providing quantitative agreement with the bulk rheology of high modulusheterogeneous materials (Evans 2004).

3.2.5. Atomic force microscopy. Dynamic AFM experiments are faced with the challenge ofdecoupling the viscoelasticity of the sample from the hydrodynamic interaction of the cantileverwith the surface. This is a partially solved problem, but research continues driven by the widerange of possible applications.

726 T A Waigh

(a)

(b)

Figure 35. (a) The rescaled indentation force plotted against the square root of the amplitudeof oscillation, and (b) the storage (E′) and loss (E′′) longitudinal modulus measured with AFM(Mahaffy et al 2000).

The question of dynamic AFM was approached by attachment of a polystyrene bead tothe cantilever (Mahaffy et al 2000). Oscillating the cantilever above a surface provided thecomplex longitudinal modulus as a function of frequency. The spherical Hertz model was usedto analyse the viscoelasticity and it is valid for adhesive forces at small indentation depths.The analysis procedure was established in the range δ/R < 0.4, where δ is the amplitude ofthe oscillations of the cantilever and R is the radius of the polystyrene bead. The complexlongitudinal modulus (E∗) was then calculated using the relationship

E∗ = fosc

δ√

δ0R, (42)

where δ0 is the initial indentation depth, and fosc is the applied force on the bead corrected forthe hydrodynamic drag force. In figure 35 the complex longitudinal modulus measured withAFM is shown as a function of frequency from the surface of a polyacrylamide gel (Mahaffyet al 2000).

AFM can also be used as dynamic mechanical testing apparatus using the power spectrumof the cantilever fluctuations (Benmouna and Johannsmann 2004). With this method theviscoelasticity of gelatine surfaces was examined with a micron sized glass sphere attached tothe cantilever.

Lubrication forces can be negligible with very soft materials (viscoelasticity dominates)(Benmouna and Johannsmann 2004). An alternative approach using an unmodified pyramidaltip has been performed with such materials (Alcaraz et al 2003). An expression is used for theapplied force (F ) on the indentation depth (δ) into a purely elastic surface.

E = 3E tan θ

4(1 − ν2)δ2, (43)

where E is the Young’s modulus, θ is the indenter angle, and ν is the Poisson ratio. Theresult was extended to produce an expression for the complex shear modulus (G∗) using the


correspondence principle. The resultant equation is:

G∗(ω) = 1 − ν

3δ0 tan θ

[F(ω)

δ(ω)− iωb(0)

], (44)

where F is the Fourier transform of the applied force, ω is the frequency, δ is the Fouriertransform of the indentation depth as a function of time, and b(0) is the drag coefficient.There continue to be some open questions concerning the role of tip surface hydrodynamicsand the role of adhesive forces in this approach. The indentation method was used to examinethe surface viscoelasticity of lung cells, but should be considered a qualitative measure until itis calibrated against a number of well understood soft condensed matter systems.

A number of different AFM techniques are being examined to study the dynamics ofsingle molecules. In a particularly interesting example, both the cantilever and surface wereoscillated during force measurement (Okajima et al 2004) facilitating the separation of thehydrodynamic effects.

3.2.6. Miscellaneous. There is a wide range of other microrheology experiments which donot fall into the aforementioned categories.

A piezorheometer has been described in a study (Constantin et al 2002) to examine thelinear rheology of surfactant micelles in the frequency range of 1 to 6 × 104 rad s−1 with asample thickness of ∼60 µm. The very small motion of the piezoelectric crystals (1 nm) usedto apply an oscillatory force to the micellar solutions, ensured that the measured viscoelasticitywas in the linear regime.

SFA has not been covered in detail in this review (Meyer et al 1998). The extremesensitivity to which SFA can probe surface forces (Israelachvili 1992), implies that thetechnique will play a major role in the development of quantitative measurements of surfaceshear moduli and reference should be made to the specialized literature.

An assortment of microrheology techniques use fluorescent tags attached to biologicalmolecules to study their motion (Le Goff et al 2002). Typically the fluctuation spectrum ofthe motion of the tags needs to be examined with a well-defined theoretical model, for thedynamics of the fluid to be understood, e.g. that of semi-flexible filaments (Morse 1998).

An active microrheology technique using electric fields to move charged colloidal particleshas been demonstrated (Mizuno et al 2001). This electrophoretic microrheology methodis probably less promising for the measurement of a general complex fluid than magnetictweezers, since it is difficult to isolate and characterize the effects of the electrostatic andelectrodynamic interactions of the sample on the probe particle motions.

The measurement of microrheology with quasi-elastic light scattering is not confined tomultiple scattering (DWS) measurements. The compliance (and thus the linear rheology)can be extracted from the opposite limit of single particle scattering with photon correlationspectroscopy (Sohn et al 2004). This microrheology DLS technique requires that theviscoelastic specimen is contrast matched with the solvent and there is a large differencein refractive index between the probe and the specimen/solvent matrix. The x-ray analogue ofthis technique provides the possibility of x-ray photon correlation spectroscopy (Dierker et al1995) microrheology of dense optically opaque complex fluids with gold probe particles.It could have a larger range of applicability than the optical technique due to the largerrange of samples providing good contrast for x-ray scattering. A further promising x-raytechnique examines the dynamics of microcrystals attached to biological molecules, providingthe measurement of milliradian rotational motion with millisecond time resolution (Sasakiet al 2001).

728 T A Waigh

4. Specific experimental systems

It is informative to consider the impact of the new microrheology techniques on specific classesof complex fluids. The complex fluids studied are predominantly aqueous, and both sampleopacity and the magnitude of the moduli that can be measured place restrictions on the materialswhich can be successfully examined. Crucially the agreement with bulk rheology is consideredin this discussion, since in many cases both the theory and the measurements in microrheologyexperiments rest on shaky foundations in this constantly developing field.

4.1. Polymers

Theoretical models have been developed hand in hand with experiment in the field of polymerrheology over the last half century. Dynamic models for the low frequency motion of flexible,semi-flexible and rod-like polymer chains are now well developed. Such theory has beenquantitatively used to understand the viscoelastic spectra (both linear and non-linear) frombulk rheological measurements. The reader is directed to the relevant books for theoreticalintroductions to the field (Edwards and Doi 1986, Rubinstein and Colby 2003).

For the experimentalist it is useful to develop an intuitive feeling for the resultof a rheological measurement with reference to some standard examples. The Zimm(unscreened hydrodynamics), Rouse (screened hydrodynamics) and semi-flexible modelsprovide predictions for the frequency dependence of shear moduli (G′, G′′), the magnitudeof the shear moduli, the compliance and the viscosity of isolated chains (Ferry 1980).Furthermore these studies have been extended by dynamic scaling theory to semi-diluteunentangled/entangled and concentrated regimes. The viscoelasticity of polyelectrolytes(Dobrynin et al 1995), associating polymers (Semenov and Rubinstein 1998) and gels(Kavanagh and Ross-Murphy 1998) are less well understood and quantitative theories fortheir dynamics are still not available.

4.1.1. Neutral polymers. Flexible hydrophilic polymers in good solvents are a standardsystem for microrheology, since in principle their viscoelasticity is well understood. A primeexample of such a material is PEO in water. The linear viscoelastic spectra from PEO showgood quantitative agreement between DWS (Mason and Weitz 1995), laser tracking (Masonet al 1997), bulk and PTM results (van Zanten et al 2004). The laser tracking microrheologyexperiments demonstrate Rouse modes at high frequencies (G′, G′′ ∝ ω1/2) for the overlappingchains (Schnurr et al 1997). The scaling law indicates that the hydrodynamics are screened atlength scales above the mesh size and is in agreement with theory (Rubinstein and Colby 2003).Recently is has been claimed that DWS has found a new, as yet unexplained, dependence ofthe zero shear rate viscosity on the polymer concentration (η ∝ c4.7) (van Zanten et al 2004).In these studies, the high frequency compliance (〈r2(t)〉) is fitted by a single relaxation timeMaxwell fluid

〈r2(t)〉 = kT

πa

(t

η+

1

G

), (45)

where kT is the thermal energy, t is the time, η is the viscosity, G is the shear modulus anda is the particle radius. Such results on PEO need to be tested in detail, since these hydrophilicpolymers are expected to provide model polymeric behaviour due to their tightly controlledpolydispersity and well quantified polymer/solvent interaction.

DWS was used to study the associating polymer rheology of hydrophobically end-cappedPEO (Lu and Solomon 2002). Good agreement was not found with the bulk rheology andthere was a large effect of the probe size on the measurements. An in-depth study of the


1E-4 1E-3 0.01 0.1

1

10

100

3/2

1/2

100418

η-η o

/η0

c[g/ml]

Figure 36. Zero shear rate viscosity data from particle-tracking microrheology experiments onflexible polyelectrolytes (maleic anhydride copolymers) at a series of different chain lengths(04, 10, 18, molecular weights are 82000, 196000, and 410000, respectively). The cross overbetween semi-dilute (∼c1/2) and entangled dynamics (∼c3/2) is shown and fair agreement isfound with the degree of chain expansion measured with small angle neutron and x-ray scatteringexperiments (Di Cola et al 2004).

effect of sample heterogeneity on the DWS results gave six possible reasons for this lack ofagreement of micro and macro results; failure of the assumption of continuum elasticity, theeffect of probe particle inertia, the compressibility of the matrix network, chain adsorptionon the surface of the probe particles, the entropic depletion of the polymer molecules, andstructural heterogeneity of the associating polymer network. The authors argue convincinglythat only the sixth point can account for the disagreement.

4.1.2. Charged polymers. Measurement of the rheology of non-associating linear flexiblepolyelectrolytes appears to be another success story of PTM. Figure 36 shows such PTMexperiments with linear flexible polyelectrolytes (maleic anhydride copolymers) which hadbeen previously examined with bulk rheological techniques (Di Cola et al 2004). Goodquantitative agreement between the specific viscosity derived from micro and macro resultswas found. The scaling laws shown are in good agreement with the theoretical predictions forthis class of charged polymers (Dobrynin et al 1995) and PTM has thus proven to be a rapidmethod for the analysis of the dynamic phase diagrams of flexible polyelectrolytes.

PTM experiments with branched flexible polyelectrolytes have also shown fair agreementbetween bulk and particle tracking data. Scaling theory (Dobrynin et al 1995) applied to x-rayand PTM data indicates that the chain backbones are almost completely extended (figure 37).Combs 16 and 13 have twice the backbone size of the other comb polymers shown in the figure,causing the large change in the entanglement concentration, and demarcating the transitionbetween screened Zimm and reptative dynamics. The number and size of the side-chainscauses a more minor change in the intrinsic viscosity with the trends expected from scalingtheory.

The large depletion forces, which would be expected to act in polystyrene sulphonatecomb/sulphonated sphere systems, do not present a barrier to micro/macro agreement. Furthertransmission DWS experiments have allowed measurement of the high frequency Maxwell-like terminal behaviour of dilute polyelectrolyte solutions up to 8 × 106 Hz (figure 38)

730 T A Waigh

24

-2

COMB 11COMB 12COMB 13COMB 14COMB 16

log

(vi

sco

sity

(P

as)

)

log (C(M))

Ce1 Ce2

Figure 37. Zero shear rate viscosity data from particle-tracking microrheology experiments onwell-defined anionically polymerized polystyrene sulphonate combs. The predictions from scalingtheory for semi-dilute (c1/2) and entangled (c3/2) solutions are shown. The vertical straightlines (ce1 and ce2) indicate the entanglement concentration marking the cross over between semi-dilute/entangled regimes.

105 106

101

102

103

104

G',G

'' (P

a)

ω (rad/sec)

G' G'' Maxwell model Fit

Figure 38. DWS data for the viscoelasticity of polystyrene sulphonate comb indicating the highfrequency Maxwell type terminal behaviour of the charged flexible comb polyelectrolytes presentedin figure 37. The contribution of the probe beads needs to be considered in very dilute solutions inwhich the sample viscoelasticity does no dominate.

(Papagiannopoulos et al 2005). This corresponds to the lowest Zimm rotational mode ofthe dynamics of the short polymeric chains.

Semi-flexible polyelectrolytes in high salt conditions have been comprehensively studied inthe form of F-actin solutions (Morse 1998, Xu et al 1998). DWS indicates that semi-flexible


modes (G′, G′′ ∝ ω3/4) are important for determining the viscoelasticity. The effects ofnetwork heterogeneity are cause for concern in microrheology experiments with semi-flexiblepolyelectrolytes and the agreement of single-particle experiments with bulk rheology is notguaranteed (Crocker et al 2000).

The area of polyelectrolyte complexation is of importance to the study of gene transfection.Oppositely charged chains (DNA) are absorbed to positively charged colloidal particles and themotion of the colloids can be studied with PTM as they traverse the cell membrane (Dawsonet al 2004).

4.1.3. Biopolymers. Actin is one of the most often studied systems in microrheologyexperiments. The motivation in this is more for its importance in cell motility than actin beingan ideal model complex fluid system. There are over 50 proteins known to associate with actin,many of which act as cross-linkers and will have a significant impact on the viscoelasticity ofthe specimens. This makes purification a veritable minefield (Bray 1992). Actin does offerlarge contour lengths and semi-flexibility, which are only available in a few other systems;DNA, viruses, F-sickle cell aggregates and self-assembled peptides (Aggeli et al 2001). Suchsamples therefore provide a significant elastic shear modulus at low polymer concentrationsand thus PTM experiments can measure G′ and G′′ simultaneously. Actin occurs in a series ofdifferent roles in living organisms. Three principle categories are structural integrity, myosin-driven motility and self-assembling treadmilling motility. Cross-linker protein almost certainlyexisted in one of the early microrheology studies (MacKintosh et al 1995) and the theoryincludes this feature of fixed entanglements. Further bulk rheology experiments revisitedthe theory in detail (Gardel et al 2004) and found agreement using samples with carefullycontrolled densities of cross-linker proteins. Most strikingly, the elastic shear modulus scaledwith polymer concentration (c) as G′ ∝ c2.2 and not c1.2 as with uncrosslinked materials. Fluidactin suspensions display better agreement with the theories of other authors (Farge and Maggs1993, Gotter et al 1996, Morse 1998).

A further novel feature of the phase behaviour of F-actin is that it can exhibit liquidcrystalline phases. Experiments have been performed to measure the microrheology aboveand below the nematic/isotropic boundary, demonstrating the viability of particle trackingtechniques in the oriented polymer phase. An elegant tracking fluorescence microscopy studyof actin filaments measured the dynamics of the end to end distance and orientation of the endsof the self-assembled fibres on time (t). Power laws of t3/4 and t1/4 were found, respectively(Le Goff et al 2002). An interesting challenge with self-assembling biopolymer systems is tostudy the microrheology around the critical micellar concentration, since the dramatic changein sample morphology has a large effect on the viscoelasticity obtained.

Single actin aggregates immersed in a network (Goodsell 1992) are an important area ofresearch, since cells are dense environments and the only study presented to date has been theforced reptation of actin filaments (Dichtl and Sackmann 2002). The gap between theory andexperiment still exists in these systems.

Type I Collagen (table jelly) was studied (Velegol and Lanni 2001) in the form of denaturedphysically cross-linked hydrogels. The heterogeneity of the gel was mapped using a laser-trapmicrorheology technique. Examination of the shear modulus at 100 nm steps across the gelprovided a range of different quantities with an order of magnitude spread in their values.Similarly, AFM has been used to map the viscoelasticity of gelatine surfaces (Benmouna andJohannsmann 2004). Collagen gels are deduced to be chemically heterogeneous and are notideal systems to study the physics of gelation. A number of developments in their chemicalanalysis would be required, such as knowledge of the charge on the chains and the numberof cross-links, to enable the gap between theory and experiment to be bridged (Dobrynin

732 T A Waigh

2.0x1013

4.0x1013

6.0x1013

8.0x1013

1.0x1014

1.2x1014

1.2

1.3

1.4

1.5

η [c

P]

c/N [1/cm3]

0.1 10.1

1

bufferc=0.12 mg/mLc=0.19 mg/mLc=0.34 mg/mLc=0.48 mg/mL

<∆x

2>

[µm

2]

lagtime [s]

(a) (b)

Figure 39. Particle tracking microrheology from the giant muscle protein titin. (a) Average MSDsat a series of titin concentrations, and (b) corresponding viscosity as a function of titin concentrationshowing the fit of the Flory/Fox equation (Di Cola 2004).

et al 2004). Collagens are however of large importance to the food industry and they play anumber of roles in the human body, with their malfunction of interest to medical science inosteoarthritis.

Gliadins are the proteins found in wheat that are responsible for bread dough elasticity.The proteins were the subject of a particle tracking study (Xu et al 2002). Progress was madein quantifying the sample heterogeneity using the histogram method.

Xanthan linear viscoelasticity was examined with DWS (Pashkovski et al 2003). Xanthanwas found to have rubber elasticity in agreement with the Mooney–Rivlin model (Ferry 1980).In the same study the shear moduli of carboxymethyl cellulose were found to demonstrate thesignature of semi-flexibility at high frequencies G′, G′′ ∝ ω3/4 (Pashkovski et al 2003).

Guar was examined with two point microrheology to demonstrate the technique with achallenging polydisperse heterogeneous system (Crocker et al 2000). Agreement was foundwith bulk rheology experiments using a newly developed theory (Levine and Lubensky 2000).

Starch dispersions with gelating agents were examined with DWS (Heinemann et al 2004).High frequency Rousse modes were observed in semi-dilute suspensions (G′, G′′ ∝ ω1/2) andthe evolution of the viscoelasticity during gelation was observed dynamically.

Aggrecan, which displays a comb comb morphology (a bottle brush of bottle brushes)(Papagiannopoulos et al 2005) was examined with PTM. Agreement was achieved betweenone- and two-particle results demonstrating that negligible sample heterogeneity occurs in thesespecimens. The zero shear rate viscosity scaled as c1.5±0.1 in common with flexible linear andcomb polyelectrolytes in the entangled regime, and Zimm modes in the shear moduli of dilutesuspensions (G′, G′′ ∝ ω2/3) were observed at high frequencies.

Titin is a giant muscle protein, which plays an important role in the passive elasticity ofstriated muscle. Although not technically a polymer (it is a single molecule with no uniquerepeating unit) it is included for continuity.

The viscosity of titin solutions was examined with PTM (figure 39) (Di Cola et al 2004).The radius of gyration calculated using the Flory–Fox equation was in agreement with thevalue measured with DLS experiments (figure 39(b)) (Higuchi et al 1993). The contributionof the semi-flexible internal modes was not observed in the viscoelastic measurements, sincethe contour length (∼1 µm) was much larger than the persistence length (∼10 nm). In contrastsmall angle neutron scattering was able to provide a clear signature of the persistence length(Di Cola et al 2004).


DNA PTM has been used to establish the size of super-coiled λ-DNA using intrinsicviscosity plots (Goodman et al 2002). There is currently a substantial use of intrinsic viscositymeasurements as a function of polymer concentration to calculate molecular sizes using bulktechniques (Kulicke and Clasen 2004) and therefore PTM experiments could have a largeimpact in this field.

4.1.4. Gels. The gel state of a complex fluid occurs in a number of different guises (Kavanaghand Ross-Murphy 1998). Principally the division exists between chemical gels in whichcovalent bonds link the sub-units and physical gels in which weaker ionic and hydrophobicinteractions are important for the association of sub-units. Gels like molecular glasses, theirsolid state analogues, present a challenging area of study due to the existence of non-ergodicdisorder. An exact definition of the gelled state is problematic due to the complex interplay ofsynerisis, heterogeneity and non-ergodicity.

DWS has examined the rheological features of gels using mechanical averaging ofcorrelation functions at different points across a polyacrylic acid gel (Nisato et al 2000).The local shear modulus was obtained as a function of the cross-linking fraction. DWS alsohas allowed the shear modulus to be extracted from the maximum MSD of probe particles(δ) in chemically cross-linked polyvinyl alcohol (PVA) gels at short times using equation (46)(Narita et al 2001).

µ = kT

6πRδ2, (46)

where R is the radius of the probe spheres. The equation is motivated by assuming the worknecessary to move a particle of radius R a distance δ in a gel (6πRδ2)µ is equal to kT . Thesestudies show that the high frequency viscoelasticity is the same for solutions and highly cross-linked PVA gels, whereas gels with low cross-linking exhibit lower storage moduli. Stronglyassociating polymers, such as hydrophobically modified PEO which form physical gels, havebeen studied using DWS (Lu and Solomon et al 2002). The single particle measure of the bulkrheology has been shown to break down in this case as discussed in section 2.2.

4.1.5. Single molecules. Single molecule experiments of polymers with well-definedhydrodynamic environments (i.e. distant from surfaces, with accurately controlled chemistriesand well-defined ionic equilibria) are still relatively few in number. The majority have beenrestricted to giant biopolymers such as titin, DNA and actin, since they allow for imagingusing optical microscopy and fluorescent probes (Dichtl and Sackmann 2002, LeDuc et al1999). Small oligomers of dextran and IG domains (sections of titin) have been examinedwith dynamic AFM near surfaces, but robust data analysis has still not been established withthese methods (section 3.2.5).

Individual titin molecules have been the subject of a number of single moleculeexperiments. Optical tweezer studies have tended to focus on equilibrium elastic propertiesas individual domains along the protein are unfolded followed by subsequent stress relaxation(Tskhovrebova et al 1997). A novel molecular combing experiment (Tskhovrebova and Trinick2001) examined the dynamics of the titin molecules and the viscoelasticity.

There are a series of other single molecule techniques, which can be used for dynamic, andconsequently rheological, measurements. These include single molecules in liquid droplets,near field scanning optical microscopy, wide field epi illumination and far field confocalmicroscopy. A good review has been written (Nie and Zare 1997).

734 T A Waigh

4.2. Colloids

Organic hard sphere colloids at high frequencies were examined using PMMA particlesmixed with probe silica spheres in DWS experiments (Sohn et al 2004). The high frequencyviscoelasticity was extracted from the correlation functions and allowed the GSE equation (15)to be tested successfully. Charged polystyrene colloids have also been studied and their gelationwas induced by controlling the ionic strength (Romer et al 2001). The buoyancy of the particleswas tuned using a mixture of H2O and D2O. The time evolution of the viscoelasticity wasexamined with DWS with 1 min resolution of the time steps.

Surfactants were examined with transmission and backscattering CCD DWS to examinethe motion of worm-like micelles (Cardinaux et al 2002). Although an accomplished technicalfeat, the data was unsatisfactory with incomplete agreement between bulk/micro viscoelasticspectra. A factor of 1.5–2 was needed to vertically rescale the curves of shear modulus againstfrequency. A single Maxwell relaxation mode was used to fit the data indicating that a singlerelaxation process controls the dynamics of these materials.

The membrane structures formed from surfactants can exhibit rich rheological behaviour(Helfer et al 2001). The in-plane and out-of-plane fluctuations of naturally occurring membranelipids have been related to the structure of the underlying actin cytoskeleton (Helfer et al 2001).Microelasticity experiments of membrane structures are not covered in this review, but theyhave had an important impact in the field of membranes, e.g. micropipette aspiration (Zhelevet al 1994).

Hexane/water emulsions of sodium dodecyl sulfate were examined with echo DWS andthe yielding of droplets was measured as a function of strain amplitude (Hebraud et al 1997).The solid/liquid shear transition of Gillette shaving cream was examined (Cohen Addad et al1998) and the process of foam coarsening during the ageing process were studied.

Inorganic silica particles have been used as probes in optical tweezer studies (Vermaet al 2000), so there is no practical barrier preventing further development of this field.Such inorganic colloidal materials are known to exhibit a range of phase behaviour. Thisincludes solid, liquid, gas, liquid-crystal, glass and gel phases, and with each there is anindividual rheological question raised (Larson et al 1999). Granular matter has been thesubject of a number of particle tracking studies using direct optical imaging. Here evidencehas been found for superdiffusive driven motion and geometry dependent caging phenomena(Choi et al 2004). However, the length scales considered with current measurements ongranular matter (spheres of ∼millimetres diameter) are typically three orders of magnitudelarger than encountered in microrheology experiments, although there is no pratical limitto a reduction in size towards the powder (micrometre-sized) state with the concomitanttechnological implications (pharmaceutical tablets, detergents, etc). There are a number ofbeautiful studies in the literature on the effects of confinement on granular particle dynamics(compare section 2.5) (Clement 1999).

4.3. Biological assemblies

Biophysics requires an understanding of the dynamics of living constructs of multi-componentcomplex fluids such as the motility of cells, the division of chromosomes, the adaptation of theextracellular matrix, and the chemotaxis of bacteria (Heidemann and Wirtz 2004). These offera whole series of challenges in both non-equilibrium thermodynamics and microrheologicalinstrumentation.

Active actin networks (Keller et al 2003) formed from mixtures of actin, myosin andATP are now being studied both theoretically and experimentally. Detailed predictions for


the rheological functions have been made and single actively transported actin filaments havebeen imaged (Humphrey et al 2002).

The viscoelasticity of the cellular cytoplasm has been approached (Bausch et al 1999).Magnetic beads are attached to the cell membrane and the viscoelasticity has been quantifiedas a function of position. The coupling of the membrane to the elasticity of the cytoskeletoncan be examined (Bausch et al 1999).

The microrheology of living cells subject to the rotational motion of ferromagnetic probeshas been studied (Fabry et al 2001). The complex modulus (G∗) was calculated from theapplied complex torque as a function of frequency (T ∗(f )) and the resultant complex angulardisplacement d∗(f )

G∗ = T ∗(f )

d∗(f)(47)

by rotating ferromagnetic particles in a magnetic field as a function of the heterodyne frequency.The samples exhibited soft glassy rheology and the shear modulus could be fitted with a modelof structural hysteretic damping.

The flagellar motor proteins that propel bacterial cells have been subject to a large numberof experiments (Berg 1993, 2003). Here the tracked particle is a living object (the bacterium),that experiences driven biased diffusion as it searches for food molecules. Poisson statisticsare observed for the probability distribution of the displacement of the particles with time andnot the Gaussian statistics found for inert probes (Crocker and Grier 1996).

A barrier to the use of probe particles in living microorganisms is the phenomenon ofphagocytosis. A number of single-cell organisms will ingest probe particles as if they were foodpackages. The tracking microrheology experiments in these cases are probing the complicatedprocesses of phagoytic cup formation and not the viscoelasticity of the unperturbed cytoplasm(Feneberg et al 2001, Ishikawa et al 2003). Such phenomena will frustrate naı̈ve studies ofthe microrheology of the components of living cells.

Fortunately, the questions concerning phagocytosis have been overcome with a number ofcell lines. Particles in these cases have been engineered to experience a less complex processof endocytosis. They are compartmentalized within a lipid membrane, but these packagesare allowed to move freely within the cell and are subjected to the representative thermalfluctuations due to the internal viscoelasticity of the cell. For example the microrheology ofliving cells has been considered using two-particle cross-correlation techniques (Lau et al2003). In another experiment, 8 nm magnetic particles were used, which formed linearaggregates in the cell allowing their rotational motion to be studied. These probes followed awell-defined process of endocytosis, facilitating the analysis (Wilhelm et al 2003).

The questions on cell/particle interaction are intimately connected with the field of genetherapy (Suh et al 2004). Here the probe particles are carriers of DNA and the question is howto move the particles across the cell wall to deliver the replacement genetic information to thecell nucleus.

AFM has been used to probe the rheology of the surfaces of a number of viscoelasticcellular materials such as lungs (Alcaraz et al 2003) and fibroblast cells (Mahaffy et al2000). More work is still needed to be certain that the measurements are characteristic ofthe specimens, and not the hydrodynamics of the cantilever or the adhesive forces with the tip.

5. Future avenues of research

Microrheological techniques need to be rigorously established with heterogeneous materialsto establish themselves alongside bulk rheometers as a range of standard characterization

736 T A Waigh

techniques. Once this is accomplished, further minimization of sample volumes should bepossible using AFM, single-fibre DWS (Popescu et al 2002) and magnetic/optical tweezers.However, the technical questions relating to these measurements are not just confined to theinstrumentation; further theoretical developments are needed to enable the experimentalistto understand what they have measured. The statistical analysis of single molecularrheological events requires thorough investigation to guide the research as the length scaleof microrheological measurements is reduced. The study of the viscoelasticity of the surfacesof complex fluids is still in it infancy and big breakthroughs are expected in this area.

The lion’s share of experiments reported to date consider the microrheology of aqueoussolutions, which are facilitated by the easy availability of electrostatically stabilized neutrallybuoyant colloidal probes. Only a small subset of the possible aqueous complex fluid systemshas been examined and a wide range of unexplored areas of soft condensed matter still exist.New probe chemistries in a series of non-aqueous solvents (preferably sterically stabilized)would greatly enhance the range of complex fluids that can be studied.

Magnetic tweezers and video particle tracking apparatus, which are both relatively cheapto construct, have as yet not been commercialized. This would cause a rapid expansion in theusage of microrheology methods.

6. Conclusions

The microrheological techniques discussed, in all their various guises, offer a series of uniqueprobes of the viscoelastic behaviour of complex fluids. Their new contributions have beenreviewed to both the measurement of the high frequency viscoelasticity and the linear rheologyof minute specimens of complex fluids in a range of confined geometries.

The future for sub-micrometre rheology is very promising. The techniques describedhere neatly complement the development of a range of single molecule experiments and wouldhelp extend these methods to provide a description of the time response of single molecules.There is an extensive overlap of microrheology methods with biophysical studies; in particular,questions such as how molecular motors dissipate energy (Howard 2001), how DNA can betransported into a cell for gene therapy (Dawson et al 2004), and the dissipative properties ofblood clots (Ryan et al 1999), would all profit from a consideration of the unified principlesof microrheology.

There are many medical and industrial applications for the rapid microanalysis ofviscoelastic samples. These require developments in colloidal synthesis to be made handin hand with experimental techniques.

Video PTM, magnetic/optical tweezers, DWS, SFA and AFM are all now well-establishedtechniques for the measurement of how a selection of complex fluids store and dissipateenergy. Currently, microrheology techniques can provide accurate measurements of the linearviscoelasticity of a number of complex fluids over a wide range of frequencies, providedthe static structure of the sample is well characterized and the chemistry of the probe/sampleinteraction is well understood (Valentine et al 2004). Extension of these methods to the routinequantification of the viscoelasticity of heterogeneous materials will be an important advance.Recent cross-correlation methods, although mathematically very sophisticated, look to be apromising solution.

Acknowledgments

Tanniemola Liverpool, Tom McLeish, Mike Evans, Alison Voice, Catherine Byrne, PeterOlmsted, Aris Papagiannopoulos, Emanuela de Cola, Edoardo de Luca, Alison Hodrien,


Manlio Tassieri, Carolina Galmes, Lisa Carrick, John Trinick, Larissa Tskhovrebova, AlastairSmith, Neil Thomson, and Rob Harrand are all thanked for useful discussions.

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